Abundant odd numbers
You are encouraged to solve this task according to the task description, using any language you may know.
An Abundant number is a number n for which the sum of divisors σ(n) > 2n,
or, equivalently, the sum of proper divisors (or aliquot sum) s(n) > n.
- E.G.
12 is abundant, it has the proper divisors 1,2,3,4 & 6 which sum to 16 ( > 12 or n);
or alternately, has the sigma sum of 1,2,3,4,6 & 12 which sum to 28 ( > 24 or 2n).
Abundant numbers are common, though even abundant numbers seem to be much more common than odd abundant numbers.
To make things more interesting, this task is specifically about finding odd abundant numbers.
- Task
- Find and display here: at least the first 25 abundant odd numbers and either their proper divisor sum or sigma sum.
- Find and display here: the one thousandth abundant odd number and either its proper divisor sum or sigma sum.
- Find and display here: the first abundant odd number greater than one billion (109) and either its proper divisor sum or sigma sum.
- Reference
American Journal of Mathematics, Vol. 35, No. 4 (Oct., 1913), pp. 413-422 - Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors (LE Dickson)
Ada
This solution uses the package Generic_Divisors from the Proper Divisors task [[1]].
<lang Ada>with Ada.Text_IO, Generic_Divisors;
procedure Odd_Abundant is
function Same(P: Positive) return Positive is (P);
package Divisor_Sum is new Generic_Divisors
(Result_Type => Natural, None => 0, One => Same, Add => "+");
function Abundant(N: Positive) return Boolean is
(Divisor_Sum.Process(N) > N);
package NIO is new Ada.Text_IO.Integer_IO(Natural);
Current: Positive := 1;
procedure Print_Abundant_Line
(Idx: Positive; N: Positive; With_Idx: Boolean:= True) is
begin
if With_Idx then
NIO.Put(Idx, 6); Ada.Text_IO.Put(" |");
else
Ada.Text_IO.Put(" *** |");
end if;
NIO.Put(N, 12); Ada.Text_IO.Put(" | ");
NIO.Put(Divisor_Sum.Process(N), 12); Ada.Text_IO.New_Line;
end Print_Abundant_Line;
begin
-- the first 25 abundant odd numbers
Ada.Text_IO.Put_Line(" index | number | proper divisor sum ");
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
for I in 1 .. 25 loop
while not Abundant(Current) loop
Current := Current + 2;
end loop;
Print_Abundant_Line(I, Current);
Current := Current + 2;
end loop;
-- the one thousandth abundant odd number
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
for I in 26 .. 1_000 loop
Current := Current + 2;
while not Abundant(Current) loop
Current := Current + 2;
end loop;
end loop;
Print_Abundant_Line(1000, Current);
-- the first abundant odd number greater than 10**9
Ada.Text_IO.Put_Line("-------+-------------+--------------------");
Current := 10**9+1;
while not Abundant(Current) loop
Current := Current + 2;
end loop;
Print_Abundant_Line(1, Current, False);
end Odd_Abundant;</lang>
- Output:
Index | Number | proper divisor sum
-------+-------------+--------------------
1 | 945 | 975
2 | 1575 | 1649
3 | 2205 | 2241
4 | 2835 | 2973
5 | 3465 | 4023
6 | 4095 | 4641
7 | 4725 | 5195
8 | 5355 | 5877
9 | 5775 | 6129
10 | 5985 | 6495
11 | 6435 | 6669
12 | 6615 | 7065
13 | 6825 | 7063
14 | 7245 | 7731
15 | 7425 | 7455
16 | 7875 | 8349
17 | 8085 | 8331
18 | 8415 | 8433
19 | 8505 | 8967
20 | 8925 | 8931
21 | 9135 | 9585
22 | 9555 | 9597
23 | 9765 | 10203
24 | 10395 | 12645
25 | 11025 | 11946
-------+-------------+--------------------
1000 | 492975 | 519361
-------+-------------+--------------------
*** | 1000000575 | 1083561009
ALGOL 68
<lang algol68>BEGIN
# find some abundant odd numbers - numbers where the sum of the proper # # divisors is bigger than the number # # itself #
# returns the sum of the proper divisors of n #
PROC divisor sum = ( INT n )INT:
BEGIN
INT sum := 1;
FOR d FROM 2 TO ENTIER sqrt( n ) DO
IF n MOD d = 0 THEN
sum +:= d;
IF INT other d := n OVER d;
other d /= d
THEN
sum +:= other d
FI
FI
OD;
sum
END # divisor sum # ;
# find numbers required by the task #
BEGIN
# first 25 odd abundant numbers #
INT odd number := 1;
INT a count := 0;
INT d sum := 0;
print( ( "The first 25 abundant odd numbers:", newline ) );
WHILE a count < 25 DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
a count +:= 1;
print( ( whole( odd number, -6 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
)
FI;
odd number +:= 2
OD;
# 1000th odd abundant number #
WHILE a count < 1 000 DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
a count := a count + 1
FI;
odd number +:= 2
OD;
print( ( "1000th abundant odd number:"
, newline
, " "
, whole( odd number - 2, 0 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
);
# first odd abundant number > one billion #
odd number := 1 000 000 001;
BOOL found := FALSE;
WHILE NOT found DO
IF ( d sum := divisor sum( odd number ) ) > odd number THEN
found := TRUE;
print( ( "First abundant odd number > 1 000 000 000:"
, newline
, " "
, whole( odd number, 0 )
, " proper divisor sum: "
, whole( d sum, 0 )
, newline
)
)
FI;
odd number +:= 2
OD
END
END</lang>
- Output:
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4641
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 6129
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
BASIC256
<lang BASIC256> numimpar = 1 contar = 0 sumaDiv = 0
function SumaDivisores(n) # Devuelve la suma de los divisores propios de n suma = 1 i = int(sqr(n))
for d = 2 to i if n % d = 0 then suma += d otroD = n \ d if otroD <> d Then suma += otroD end if Next d Return suma End Function
- Encontrar los números requeridos por la tarea:
- primeros 25 números abundantes impares
Print "Los primeros 25 números impares abundantes:" While contar < 25 sumaDiv = SumaDivisores(numimpar) If sumaDiv > numimpar Then contar += 1 Print numimpar & " suma divisoria adecuada: " & sumaDiv End If numimpar += 2 End While
- 1000er número impar abundante
While contar < 1000 sumaDiv = SumaDivisores(numimpar) print sumaDiv & " " & contar If sumaDiv > numimpar Then contar += 1 numimpar += 2 End While Print Chr(10) & "1000º número impar abundante:" Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv
- primer número impar abundante > mil millones (millardo)
numimpar = 1000000001 encontrado = False While Not encontrado sumaDiv = SumaDivisores(numimpar) If sumaDiv > numimpar Then encontrado = True Print Chr(10) & "Primer número impar abundante > 1 000 000 000:" Print " " & numimpar & " suma divisoria adecuada: " & sumaDiv End If numimpar += 2 End While End </lang>
C
<lang c>#include <stdio.h>
- include <math.h>
// The following function is for odd numbers ONLY // Please use "for (unsigned i = 2, j; i*i <= n; i ++)" for even and odd numbers unsigned sum_proper_divisors(const unsigned n) {
unsigned sum = 1; for (unsigned i = 3, j; i < sqrt(n)+1; i += 2) if (n % i == 0) sum += i + (i == (j = n / i) ? 0 : j); return sum;
}
int main(int argc, char const *argv[]) {
unsigned n, c;
for (n = 1, c = 0; c < 25; n += 2) if (n < sum_proper_divisors(n)) printf("%u: %u\n", ++c, n);
for ( ; c < 1000; n += 2) if (n < sum_proper_divisors(n)) c ++;
printf("\nThe one thousandth abundant odd number is: %u\n", n);
for (n = 1000000001 ;; n += 2) if (n < sum_proper_divisors(n)) break;
printf("The first abundant odd number above one billion is: %u\n", n);
return 0;
}</lang>
- Output:
1: 945 2: 1575 3: 2205 4: 2835 5: 3465 6: 4095 7: 4725 8: 5355 9: 5775 10: 5985 11: 6435 12: 6615 13: 6825 14: 7245 15: 7425 16: 7875 17: 8085 18: 8415 19: 8505 20: 8925 21: 9135 22: 9555 23: 9765 24: 10395 25: 11025 The one thousandth abundant odd number is: 492977 The first abundant odd number above one billion is: 1000000575
C++
<lang cpp>#include <algorithm>
- include <iostream>
- include <numeric>
- include <sstream>
- include <vector>
std::vector<int> divisors(int n) {
std::vector<int> divs{ 1 };
std::vector<int> divs2;
for (int i = 2; i*i <= n; i++) {
if (n%i == 0) {
int j = n / i;
divs.push_back(i);
if (i != j) {
divs2.push_back(j);
}
}
}
std::copy(divs2.crbegin(), divs2.crend(), std::back_inserter(divs));
return divs;
}
int sum(const std::vector<int>& divs) {
return std::accumulate(divs.cbegin(), divs.cend(), 0);
}
std::string sumStr(const std::vector<int>& divs) {
auto it = divs.cbegin(); auto end = divs.cend(); std::stringstream ss;
if (it != end) {
ss << *it;
it = std::next(it);
}
while (it != end) {
ss << " + " << *it;
it = std::next(it);
}
return ss.str();
}
int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) {
int count = countFrom;
int n = searchFrom;
for (; count < countTo; n += 2) {
auto divs = divisors(n);
int tot = sum(divs);
if (tot > n) {
count++;
if (printOne && count < countTo) {
continue;
}
auto s = sumStr(divs);
if (printOne) {
printf("%d < %s = %d\n", n, s.c_str(), tot);
} else {
printf("%2d. %5d < %s = %d\n", count, n, s.c_str(), tot);
}
}
}
return n;
}
int main() {
using namespace std;
const int max = 25; cout << "The first " << max << " abundant odd numbers are:\n"; int n = abundantOdd(1, 0, 25, false);
cout << "\nThe one thousandth abundant odd number is:\n"; abundantOdd(n, 25, 1000, true);
cout << "\nThe first abundant odd number above one billion is:\n"; abundantOdd(1e9 + 1, 0, 1, true);
return 0;
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
C#
<lang csharp>using static System.Console; using System.Collections.Generic; using System.Linq;
public static class AbundantOddNumbers {
public static void Main() {
WriteLine("First 25 abundant odd numbers:");
foreach (var x in AbundantNumbers().Take(25)) WriteLine(x.Format());
WriteLine();
WriteLine($"The 1000th abundant odd number: {AbundantNumbers().ElementAt(999).Format()}");
WriteLine();
WriteLine($"First abundant odd number > 1b: {AbundantNumbers(1_000_000_001).First().Format()}");
}
static IEnumerable<(int n, int sum)> AbundantNumbers(int start = 3) =>
start.UpBy(2).Select(n => (n, sum: n.DivisorSum())).Where(x => x.sum > x.n);
static int DivisorSum(this int n) => 3.UpBy(2).TakeWhile(i => i * i <= n).Where(i => n % i == 0)
.Select(i => (a:i, b:n/i)).Sum(p => p.a == p.b ? p.a : p.a + p.b) + 1;
static IEnumerable<int> UpBy(this int n, int step) {
for (int i = n; ; i+=step) yield return i;
}
static string Format(this (int n, int sum) pair) => $"{pair.n:N0} with sum {pair.sum:N0}";
}</lang>
- Output:
First 25 abundant odd numbers: 945 with sum 975 1,575 with sum 1,649 2,205 with sum 2,241 2,835 with sum 2,973 3,465 with sum 4,023 4,095 with sum 4,641 4,725 with sum 5,195 5,355 with sum 5,877 5,775 with sum 6,129 5,985 with sum 6,495 6,435 with sum 6,669 6,615 with sum 7,065 6,825 with sum 7,063 7,245 with sum 7,731 7,425 with sum 7,455 7,875 with sum 8,349 8,085 with sum 8,331 8,415 with sum 8,433 8,505 with sum 8,967 8,925 with sum 8,931 9,135 with sum 9,585 9,555 with sum 9,597 9,765 with sum 10,203 10,395 with sum 12,645 11,025 with sum 11,946 The 1000th abundant odd number: 492,975 with sum 519,361 First abundant odd number > 1b: 1,000,000,575 with sum 1,083,561,009
D
<lang d>import std.stdio;
int[] divisors(int n) {
import std.range;
int[] divs = [1]; int[] divs2;
for (int i = 2; i * i <= n; i++) {
if (n % i == 0) {
int j = n / i;
divs ~= i;
if (i != j) {
divs2 ~= j;
}
}
}
divs ~= retro(divs2).array;
return divs;
}
int abundantOdd(int searchFrom, int countFrom, int countTo, bool printOne) {
import std.algorithm.iteration; import std.array; import std.conv;
int count = countFrom;
int n = searchFrom;
for (; count < countTo; n += 2) {
auto divs = divisors(n);
int tot = sum(divs);
if (tot > n) {
count++;
if (printOne && count < countTo) {
continue;
}
auto s = divs.map!(to!string).join(" + ");
if (printOne) {
writefln("%d < %s = %d", n, s, tot);
} else {
writefln("%2d. %5d < %s = %d", count, n, s, tot);
}
}
}
return n;
}
void main() {
const int max = 25;
writefln("The first %d abundant odd numbers are:", max);
int n = abundantOdd(1, 0, 25, false);
writeln("\nThe one thousandth abundant odd number is:");
abundantOdd(n, 25, 1000, true);
writeln("\nThe first abundant odd number above one billion is:");
abundantOdd(cast(int)(1e9 + 1), 0, 1, true);
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Factor
<lang factor>USING: arrays formatting io kernel lists lists.lazy math math.primes.factors sequences tools.memory.private ; IN: rosetta-code.abundant-odd-numbers
- σ ( n -- sum ) divisors sum ;
- abundant? ( n -- ? ) [ σ ] [ 2 * ] bi > ;
- abundant-odds-from ( n -- list )
dup even? [ 1 + ] when [ 2 + ] lfrom-by [ abundant? ] lfilter ;
- first25 ( -- seq ) 25 1 abundant-odds-from ltake list>array ;
- 1,000th ( -- n ) 1 abundant-odds-from 999 [ cdr ] times car ;
- first>10^9 ( -- n ) 1,000,000,001 abundant-odds-from car ;
GENERIC: show ( obj -- ) M: integer show dup σ [ commas ] bi@ "%-6s σ = %s\n" printf ; M: array show [ show ] each ;
- abundant-odd-numbers-demo ( -- )
first25 "First 25 abundant odd numbers:" 1,000th "1,000th abundant odd number:" first>10^9 "First abundant odd number > one billion:" [ print show nl ] 2tri@ ;
MAIN: abundant-odd-numbers-demo</lang>
- Output:
First 25 abundant odd numbers: 945 σ = 1,920 1,575 σ = 3,224 2,205 σ = 4,446 2,835 σ = 5,808 3,465 σ = 7,488 4,095 σ = 8,736 4,725 σ = 9,920 5,355 σ = 11,232 5,775 σ = 11,904 5,985 σ = 12,480 6,435 σ = 13,104 6,615 σ = 13,680 6,825 σ = 13,888 7,245 σ = 14,976 7,425 σ = 14,880 7,875 σ = 16,224 8,085 σ = 16,416 8,415 σ = 16,848 8,505 σ = 17,472 8,925 σ = 17,856 9,135 σ = 18,720 9,555 σ = 19,152 9,765 σ = 19,968 10,395 σ = 23,040 11,025 σ = 22,971 1,000th abundant odd number: 492,975 σ = 1,012,336 First abundant odd number > one billion: 1,000,000,575 σ = 2,083,561,584
FreeBASIC
<lang freebasic> Declare Function SumaDivisores(n As Integer) As Integer
Dim numimpar As Integer = 1 Dim contar As Integer = 0 Dim sumaDiv As Integer = 0
Function SumaDivisores(n As Integer) As Integer
' Devuelve la suma de los divisores propios de n
Dim suma As Integer = 1
Dim As Integer d, otroD
For d = 2 To Cint(Sqr(n))
If n Mod d = 0 Then
suma += d
otroD = n \ d
If otroD <> d Then suma += otroD
End If
Next d
Return suma
End Function
' Encontrar los números requeridos por la tarea:
' primeros 25 números abundantes impares Print "Los primeros 25 números impares abundantes:" Do While contar < 25
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
contar += 1
Print using "######"; numimpar;
Print " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
Loop
' 1000er número impar abundante Do While contar < 1000
sumaDiv = SumaDivisores(numimpar) If sumaDiv > numimpar Then contar += 1 numimpar += 2
Loop Print Chr(10) & "1000º número impar abundante:" Print " " & (numimpar - 2) & " suma divisoria adecuada: " & sumaDiv
' primer número impar abundante > mil millones (millardo) numimpar = 1000000001 Dim encontrado As Boolean = False Do While Not encontrado
sumaDiv = SumaDivisores(numimpar)
If sumaDiv > numimpar Then
encontrado = True
Print Chr(10) & "Primer número impar abundante > 1 000 000 000:"
Print " " & numimpar & " suma divisoria adecuada: " & sumaDiv
End If
numimpar += 2
Loop End </lang>
- Output:
Los primeros 25 números impares abundantes:
945 suma divisoria adecuada: 975
1575 suma divisoria adecuada: 1649
2205 suma divisoria adecuada: 2241
2835 suma divisoria adecuada: 2973
3465 suma divisoria adecuada: 4023
4095 suma divisoria adecuada: 4641
4725 suma divisoria adecuada: 5195
5355 suma divisoria adecuada: 5877
5775 suma divisoria adecuada: 6129
5985 suma divisoria adecuada: 6495
6435 suma divisoria adecuada: 6669
6615 suma divisoria adecuada: 7065
6825 suma divisoria adecuada: 7063
7245 suma divisoria adecuada: 7731
7425 suma divisoria adecuada: 7455
7875 suma divisoria adecuada: 8349
8085 suma divisoria adecuada: 8331
8415 suma divisoria adecuada: 8433
8505 suma divisoria adecuada: 8967
8925 suma divisoria adecuada: 8931
9135 suma divisoria adecuada: 9585
9555 suma divisoria adecuada: 9597
9765 suma divisoria adecuada: 10203
10395 suma divisoria adecuada: 12645
11025 suma divisoria adecuada: 11946
1000º número impar abundante:
492975 suma divisoria adecuada: 519361
Primer número impar abundante > 1 000 000 000:
1000000575 suma divisoria adecuada: 1083561009
Go
<lang go>package main
import (
"fmt" "strconv"
)
func divisors(n int) []int {
divs := []int{1}
divs2 := []int{}
for i := 2; i*i <= n; i++ {
if n%i == 0 {
j := n / i
divs = append(divs, i)
if i != j {
divs2 = append(divs2, j)
}
}
}
for i := len(divs2) - 1; i >= 0; i-- {
divs = append(divs, divs2[i])
}
return divs
}
func sum(divs []int) int {
tot := 0
for _, div := range divs {
tot += div
}
return tot
}
func sumStr(divs []int) string {
s := ""
for _, div := range divs {
s += strconv.Itoa(div) + " + "
}
return s[0 : len(s)-3]
}
func abundantOdd(searchFrom, countFrom, countTo int, printOne bool) int {
count := countFrom
n := searchFrom
for ; count < countTo; n += 2 {
divs := divisors(n)
if tot := sum(divs); tot > n {
count++
if printOne && count < countTo {
continue
}
s := sumStr(divs)
if !printOne {
fmt.Printf("%2d. %5d < %s = %d\n", count, n, s, tot)
} else {
fmt.Printf("%d < %s = %d\n", n, s, tot)
}
}
}
return n
}
func main() {
const max = 25
fmt.Println("The first", max, "abundant odd numbers are:")
n := abundantOdd(1, 0, 25, false)
fmt.Println("\nThe one thousandth abundant odd number is:")
abundantOdd(n, 25, 1000, true)
fmt.Println("\nThe first abundant odd number above one billion is:")
abundantOdd(1e9+1, 0, 1, true)
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Haskell
<lang Haskell>import Data.List (nub)
divisorSum :: Integral a => a -> a divisorSum n =
sum . map (\i -> sum $ nub [i, n `quot` i]) . filter ((== 0) . (n `rem`)) $ takeWhile ((<= n) . (^ 2)) [1 ..]
oddAbundants :: Integral a => a -> [(a, a)] oddAbundants n =
[ (i, divisorSum i) | i <- [n ..], odd i, divisorSum i > i * 2 ]
printAbundant :: (Int, Int) -> IO () printAbundant (n, s) =
putStrLn $ show n ++ " with " ++ show s ++ " as the sum of all proper divisors."
main :: IO () main = do
putStrLn "The first 25 odd abundant numbers are:" mapM_ printAbundant . take 25 $ oddAbundants 1 putStrLn "The 1000th odd abundant number is:" printAbundant $ oddAbundants 1 !! 1000 putStrLn "The first odd abundant number above 1000000000 is:" printAbundant . head . oddAbundants $ 10 ^ 9</lang>
- Output:
The first 25 odd abundant numbers are: 945 with 1920 as the sum of all proper divisors. 1575 with 3224 as the sum of all proper divisors. 2205 with 4446 as the sum of all proper divisors. 2835 with 5808 as the sum of all proper divisors. 3465 with 7488 as the sum of all proper divisors. 4095 with 8736 as the sum of all proper divisors. 4725 with 9920 as the sum of all proper divisors. 5355 with 11232 as the sum of all proper divisors. 5775 with 11904 as the sum of all proper divisors. 5985 with 12480 as the sum of all proper divisors. 6435 with 13104 as the sum of all proper divisors. 6615 with 13680 as the sum of all proper divisors. 6825 with 13888 as the sum of all proper divisors. 7245 with 14976 as the sum of all proper divisors. 7425 with 14880 as the sum of all proper divisors. 7875 with 16224 as the sum of all proper divisors. 8085 with 16416 as the sum of all proper divisors. 8415 with 16848 as the sum of all proper divisors. 8505 with 17472 as the sum of all proper divisors. 8925 with 17856 as the sum of all proper divisors. 9135 with 18720 as the sum of all proper divisors. 9555 with 19152 as the sum of all proper divisors. 9765 with 19968 as the sum of all proper divisors. 10395 with 23040 as the sum of all proper divisors. 11025 with 22971 as the sum of all proper divisors. The 1000th odd abundant number is: 493185 with 1017792 as the sum of all proper divisors. The first odd abundant number above 1000000000 is: 1000000575 with 2083561584 as the sum of all proper divisors.
J
NB. https://www.math.upenn.edu/~deturck/m170/wk3/lecture/sumdiv.html
s=: ([: */ [: ((<:@:(^ >:)/) % <:@:{.) __&q:)&>
assert 6045 -: s 1800
aliquot_sum=: -~ s
abundant=: < aliquot_sum
Filter=: (#~`)(`:6)
A=: abundant Filter 1 2 p. i. 260000 NB. a batch of abundant odd numbers
# A NB. more than 1000, it's enough.
1054
NB. the first odd abundant numbers
(,: aliquot_sum) 26 {. A
945 1575 2205 2835 3465 4095 4725 5355 5775 5985 6435 6615 6825 7245 7425 7875 8085 8415 8505 8925 9135 9555 9765 10395 11025 11655
975 1649 2241 2973 4023 4641 5195 5877 6129 6495 6669 7065 7063 7731 7455 8349 8331 8433 8967 8931 9585 9597 10203 12645 11946 12057
NB. the one thousandth abundant odd number
(,: aliquot_sum) 999 { A
492975
519361
k=: adverb def '1000 * m'
1x k k k
1000000000
abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575 1000001475 1000001625 1000001835 1000002465 1000003095 1000003725 1000004355 1000004775 1000004985 1000005435 1000005615 1000005825 1000006245 1000006425 1000006875 1000007505 1000008765 1000009395 1000010025 1000010655 1000011285 1000011705 100...
(,: aliquot_sum) {. abundant Filter (1x k k k) + 1 2x p. i. 10x k
1000000575
1083561009
Julia
<lang julia>using Primes
function propfact(n)
f = [one(n)]
for (p, x) in factor(n)
f = reduce(vcat, [f*p^i for i in 1:x], init=f)
end
pop!(f)
f
end
isabundant(n) = sum(propfact(n)) > n prettyprintfactors(n) = (a = propfact(n); println("$n has proper divisors $a, these sum to $(sum(a))."))
function oddabundantsfrom(startingint, needed, nprint=0)
n = isodd(startingint) ? startingint : startingint + 1
count = 0
while count < needed
if isabundant(n)
if nprint == 0
prettyprintfactors(n)
elseif nprint == count + 1
prettyprintfactors(n)
break
end
count += 1
end
n += 2
end
end
println("First 25 abundant odd numbers:") oddabundantsfrom(2, 25)
println("The thousandth abundant odd number:") oddabundantsfrom(2, 1001, 1000)
println("The first abundant odd number greater than one billion:") oddabundantsfrom(1000000000, 1)
</lang>
- Output:
First 25 abundant odd numbers: 945 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315], these sum to 975. 1575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525], these sum to 1649. 2205 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 49, 147, 441, 245, 735], these sum to 2241. 2835 has proper divisors [1, 3, 9, 27, 81, 5, 15, 45, 135, 405, 7, 21, 63, 189, 567, 35, 105, 315, 945], these sum to 2973. 3465 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 11, 33, 99, 55, 165, 495, 77, 231, 693, 385, 1155], these sum to 4023. 4095 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 13, 39, 117, 65, 195, 585, 91, 273, 819, 455, 1365], these sum to 4641. 4725 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 7, 21, 63, 189, 35, 105, 315, 945, 175, 525, 1575], these sum to 5195. 5355 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 17, 51, 153, 85, 255, 765, 119, 357, 1071, 595, 1785], these sum to 5877. 5775 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 11, 33, 55, 165, 275, 825, 77, 231, 385, 1155, 1925], these sum to 6129. 5985 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 19, 57, 171, 95, 285, 855, 133, 399, 1197, 665, 1995], these sum to 6495. 6435 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 13, 39, 117, 65, 195, 585, 143, 429, 1287, 715, 2145], these sum to 6669. 6615 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 49, 147, 441, 1323, 245, 735, 2205], these sum to 7065. 6825 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 13, 39, 65, 195, 325, 975, 91, 273, 455, 1365, 2275], these sum to 7063. 7245 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 23, 69, 207, 115, 345, 1035, 161, 483, 1449, 805, 2415], these sum to 7731. 7425 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 25, 75, 225, 675, 11, 33, 99, 297, 55, 165, 495, 1485, 275, 825, 2475], these sum to 7455. 7875 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 125, 375, 1125, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 875, 2625], these sum to 8349. 8085 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 11, 33, 55, 165, 77, 231, 385, 1155, 539, 1617, 2695], these sum to 8331. 8415 has proper divisors [1, 3, 9, 5, 15, 45, 11, 33, 99, 55, 165, 495, 17, 51, 153, 85, 255, 765, 187, 561, 1683, 935, 2805], these sum to 8433. 8505 has proper divisors [1, 3, 9, 27, 81, 243, 5, 15, 45, 135, 405, 1215, 7, 21, 63, 189, 567, 1701, 35, 105, 315, 945, 2835], these sum to 8967. 8925 has proper divisors [1, 3, 5, 15, 25, 75, 7, 21, 35, 105, 175, 525, 17, 51, 85, 255, 425, 1275, 119, 357, 595, 1785, 2975], these sum to 8931. 9135 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 29, 87, 261, 145, 435, 1305, 203, 609, 1827, 1015, 3045], these sum to 9585. 9555 has proper divisors [1, 3, 5, 15, 7, 21, 35, 105, 49, 147, 245, 735, 13, 39, 65, 195, 91, 273, 455, 1365, 637, 1911, 3185], these sum to 9597. 9765 has proper divisors [1, 3, 9, 5, 15, 45, 7, 21, 63, 35, 105, 315, 31, 93, 279, 155, 465, 1395, 217, 651, 1953, 1085, 3255], these sum to 10203. 10395 has proper divisors [1, 3, 9, 27, 5, 15, 45, 135, 7, 21, 63, 189, 35, 105, 315, 945, 11, 33, 99, 297, 55, 165, 495, 1485, 77, 231, 693, 2079, 385, 1155, 3465], these sum to 12645. 11025 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675], these sum to 11946. The thousandth abundant odd number: 492975 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 313, 939, 2817, 1565, 4695, 14085, 7825, 23475, 70425, 2191, 6573, 19719, 10955, 32865, 98595, 54775, 164325], these sum to 519361. The first abundant odd number greater than one billion: 1000000575 has proper divisors [1, 3, 9, 5, 15, 45, 25, 75, 225, 7, 21, 63, 35, 105, 315, 175, 525, 1575, 49, 147, 441, 245, 735, 2205, 1225, 3675, 11025, 90703, 272109, 816327, 453515, 1360545, 4081635, 2267575, 6802725, 20408175, 634921, 1904763, 5714289, 3174605, 9523815, 28571445, 15873025, 47619075, 142857225, 4444447, 13333341, 40000023, 22222235, 66666705, 200000115, 111111175, 333333525], these sum to 1083561009.
Kotlin
<lang scala>fun divisors(n: Int): List<Int> {
val divs = mutableListOf(1) val divs2 = mutableListOf<Int>()
var i = 2
while (i * i <= n) {
if (n % i == 0) {
val j = n / i
divs.add(i)
if (i != j) {
divs2.add(j)
}
}
i++
}
divs.addAll(divs2.reversed())
return divs
}
fun abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int {
var count = countFrom var n = searchFrom
while (count < countTo) {
val divs = divisors(n)
val tot = divs.sum()
if (tot > n) {
count++
if (!printOne || count >= countTo) {
val s = divs.joinToString(" + ")
if (printOne) {
println("$n < $s = $tot")
} else {
println("%2d. %5d < %s = %d".format(count, n, s, tot))
}
}
}
n += 2 }
return n
}
fun main() {
val max = 25
println("The first $max abundant odd numbers are:")
val n = abundantOdd(1, 0, 25, false)
println("\nThe one thousandth abundant odd number is:")
abundantOdd(n, 25, 1000, true)
println("\nThe first abundant odd number above one billion is:")
abundantOdd((1e9 + 1).toInt(), 0, 1, true)
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Perl
<lang perl>use strict; use warnings; use feature 'say'; use ntheory qw/divisor_sum divisors/;
sub odd_abundants {
my($start,$count) = @_;
my $n = int(( $start + 2 ) / 3);
$n += 1 if 0 == $n % 2;
$n *= 3;
my @out;
while (@out < $count) {
$n += 6;
next unless (my $ds = divisor_sum($n)) > 2*$n;
my @d = divisors($n);
push @out, sprintf "%6d: divisor sum: %s = %d", $n, join(' + ', @d[0..@d-2]), $ds-$n;
}
@out;
}
say 'First 25 abundant odd numbers:'; say for odd_abundants(1, 25); say "\nOne thousandth abundant odd number:\n", (odd_abundants(1, 1000))[999]; say "\nFirst abundant odd number above one billion:\n", odd_abundants(999_999_999, 1);</lang>
- Output:
First 25 abundant odd numbers: 945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 One thousandth abundant odd number: 492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 First abundant odd number above one billion: 1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Perl 6
<lang perl6>sub odd-abundant (\x) {
my @l = x.is-prime ?? 1 !! flat
1, (3 .. x.sqrt.floor).map: -> \d {
next unless d +& 1;
my \y = x div d;
next if y * d !== x;
d !== y ?? (d, y) !! d
};
@l.sum > x ?? @l.sort !! Empty;
}
sub odd-abundants (Int :$start-at is copy) {
$start-at = ( $start-at + 2 ) div 3;
$start-at += $start-at %% 2;
$start-at *= 3;
($start-at, *+6 ... *).hyper.map: {
next unless my $oa = .&odd-abundant;
sprintf "%6d: divisor sum: {$oa.join: ' + '} = {$oa.sum}", $_
}
}
put 'First 25 abundant odd numbers:'; .put for odd-abundants( :start-at(1) )[^25];
put "\nOne thousandth abundant odd number:\n" ~ odd-abundants( :start-at(1) )[999] ~
"\n\nFirst abundant odd number above one billion:\n" ~ odd-abundants( :start-at(1_000_000_000) ).head;</lang>
- Output:
First 25 abundant odd numbers: 945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 One thousandth abundant odd number: 492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 First abundant odd number above one billion: 1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Phix
<lang Phix>function abundantOdd(integer n, done, lim, bool printAll)
while done<lim do
atom tot = sum(factors(n,-1))
if tot>n then
done += 1
if printAll or done=lim then
string ln = iff(printAll?sprintf("%2d. ",done):"")
printf(1,"%s%,6d (proper sum:%,d)\n",{ln,n,tot})
end if
end if
n += 2
end while
printf(1,"\n")
return n
end function printf(1,"The first 25 abundant odd numbers are:\n") integer n = abundantOdd(1, 0, 25, true) printf(1,"The one thousandth abundant odd number is:") {} = abundantOdd(n, 25, 1000, false) printf(1,"The first abundant odd number above one billion is:") {} = abundantOdd(1e9+1, 0, 1, false)</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 (proper sum:975) 2. 1,575 (proper sum:1,649) 3. 2,205 (proper sum:2,241) 4. 2,835 (proper sum:2,973) 5. 3,465 (proper sum:4,023) 6. 4,095 (proper sum:4,641) 7. 4,725 (proper sum:5,195) 8. 5,355 (proper sum:5,877) 9. 5,775 (proper sum:6,129) 10. 5,985 (proper sum:6,495) 11. 6,435 (proper sum:6,669) 12. 6,615 (proper sum:7,065) 13. 6,825 (proper sum:7,063) 14. 7,245 (proper sum:7,731) 15. 7,425 (proper sum:7,455) 16. 7,875 (proper sum:8,349) 17. 8,085 (proper sum:8,331) 18. 8,415 (proper sum:8,433) 19. 8,505 (proper sum:8,967) 20. 8,925 (proper sum:8,931) 21. 9,135 (proper sum:9,585) 22. 9,555 (proper sum:9,597) 23. 9,765 (proper sum:10,203) 24. 10,395 (proper sum:12,645) 25. 11,025 (proper sum:11,946) The one thousandth abundant odd number is:492,975 (proper sum:519,361) The first abundant odd number above one billion is:1,000,000,575 (proper sum:1,083,561,009)
PicoLisp
<lang PicoLisp>(de accud (Var Key)
(if (assoc Key (val Var))
(con @ (inc (cdr @)))
(push Var (cons Key 1)) )
Key )
(de **sum (L)
(let S 1
(for I (cdr L)
(inc 'S (** (car L) I)) )
S ) )
(de factor-sum (N)
(if (=1 N)
0
(let
(R NIL
D 2
L (1 2 2 . (4 2 4 2 4 6 2 6 .))
M (sqrt N)
N1 N
S 1 )
(while (>= M D)
(if (=0 (% N1 D))
(setq M
(sqrt (setq N1 (/ N1 (accud 'R D)))) )
(inc 'D (pop 'L)) ) )
(accud 'R N1)
(for I R
(setq S (* S (**sum I))) )
(- S N) ) ) )
(de factor-list NIL
(let (N 1 C 0)
(make
(loop
(when (> (setq @@ (factor-sum N)) N)
(link (cons N @@))
(inc 'C) )
(inc 'N 2)
(T (= C 1000)) ) ) ) )
(let L (factor-list)
(for N 25
(println N (++ L)) )
(println 1000 (last L))
(println
'****
1000000575
(factor-sum 1000000575) ) )</lang>
- Output:
1 (945 . 975) 2 (1575 . 1649) 3 (2205 . 2241) 4 (2835 . 2973) 5 (3465 . 4023) 6 (4095 . 4641) 7 (4725 . 5195) 8 (5355 . 5877) 9 (5775 . 6129) 10 (5985 . 6495) 11 (6435 . 6669) 12 (6615 . 7065) 13 (6825 . 7063) 14 (7245 . 7731) 15 (7425 . 7455) 16 (7875 . 8349) 17 (8085 . 8331) 18 (8415 . 8433) 19 (8505 . 8967) 20 (8925 . 8931) 21 (9135 . 9585) 22 (9555 . 9597) 23 (9765 . 10203) 24 (10395 . 12645) 25 (11025 . 11946) 1000 (492975 . 519361) **** 1000000575 1083561009
PureBasic
<lang PureBasic>NewList l_sum.i()
Procedure.i sum_proper_divisors(n.i)
Define.i sum, i=3, j
Shared l_sum()
AddElement(l_sum())
l_sum()=1
While i<Sqr(n)+1
If n%i=0
sum+i
AddElement(l_sum())
l_sum()=i
j=n/i
If i<>j
sum+j
AddElement(l_sum())
l_sum()=j
EndIf
EndIf
i+2
Wend
ProcedureReturn sum+1
EndProcedure
If OpenConsole("Abundant_odd_numbers")
Define.i n, c, s n=1 c=0 While c<25 ClearList(l_sum()) s=sum_proper_divisors(n) If n"+RSet(Str(s),6)) ForEach l_sum() If ListIndex(l_sum())=0 Print(" = ") Else Print("+") EndIf Print(Str(l_sum())) Next PrintN("") EndIf n+2 Wend
n-2
While c<1000
s=sum_proper_divisors(n+2)
c+Bool(n<s)
n+2
Wend
PrintN(~"\nThe one thousandth abundant odd number is: "+Str(n)+
~"\n\tand the proper divisor sum is: "+Str(s))
n=1000000001-2
Repeat
n+2
s=sum_proper_divisors(n)
Until n<s
PrintN("The first abundant odd number above one billion is: "+Str(n)+
~"\n\tand the proper divisor sum is: "+Str(s))
Input()
EndIf</lang>
- Output:
1: 945 -> 975 = 1+3+5+7+9+15+21+27+35+45+63+105+135+189+315 2: 1575 -> 1649 = 1+3+5+7+9+15+21+25+35+45+63+75+105+175+225+315+525 3: 2205 -> 2241 = 1+3+5+7+9+15+21+35+45+49+63+105+147+245+315+441+735 4: 2835 -> 2973 = 1+3+5+7+9+15+21+27+35+45+63+81+105+135+189+315+405+567+945 5: 3465 -> 4023 = 1+3+5+7+9+11+15+21+33+35+45+55+63+77+99+105+165+231+315+385+495+693+1155 6: 4095 -> 4641 = 1+3+5+7+9+13+15+21+35+39+45+63+65+91+105+117+195+273+315+455+585+819+1365 7: 4725 -> 5195 = 1+3+5+7+9+15+21+25+27+35+45+63+75+105+135+175+189+225+315+525+675+945+1575 8: 5355 -> 5877 = 1+3+5+7+9+15+17+21+35+45+51+63+85+105+119+153+255+315+357+595+765+1071+1785 9: 5775 -> 6129 = 1+3+5+7+11+15+21+25+33+35+55+75+77+105+165+175+231+275+385+525+825+1155+1925 10: 5985 -> 6495 = 1+3+5+7+9+15+19+21+35+45+57+63+95+105+133+171+285+315+399+665+855+1197+1995 11: 6435 -> 6669 = 1+3+5+9+11+13+15+33+39+45+55+65+99+117+143+165+195+429+495+585+715+1287+2145 12: 6615 -> 7065 = 1+3+5+7+9+15+21+27+35+45+49+63+105+135+147+189+245+315+441+735+945+1323+2205 13: 6825 -> 7063 = 1+3+5+7+13+15+21+25+35+39+65+75+91+105+175+195+273+325+455+525+975+1365+2275 14: 7245 -> 7731 = 1+3+5+7+9+15+21+23+35+45+63+69+105+115+161+207+315+345+483+805+1035+1449+2415 15: 7425 -> 7455 = 1+3+5+9+11+15+25+27+33+45+55+75+99+135+165+225+275+297+495+675+825+1485+2475 16: 7875 -> 8349 = 1+3+5+7+9+15+21+25+35+45+63+75+105+125+175+225+315+375+525+875+1125+1575+2625 17: 8085 -> 8331 = 1+3+5+7+11+15+21+33+35+49+55+77+105+147+165+231+245+385+539+735+1155+1617+2695 18: 8415 -> 8433 = 1+3+5+9+11+15+17+33+45+51+55+85+99+153+165+187+255+495+561+765+935+1683+2805 19: 8505 -> 8967 = 1+3+5+7+9+15+21+27+35+45+63+81+105+135+189+243+315+405+567+945+1215+1701+2835 20: 8925 -> 8931 = 1+3+5+7+15+17+21+25+35+51+75+85+105+119+175+255+357+425+525+595+1275+1785+2975 21: 9135 -> 9585 = 1+3+5+7+9+15+21+29+35+45+63+87+105+145+203+261+315+435+609+1015+1305+1827+3045 22: 9555 -> 9597 = 1+3+5+7+13+15+21+35+39+49+65+91+105+147+195+245+273+455+637+735+1365+1911+3185 23: 9765 -> 10203 = 1+3+5+7+9+15+21+31+35+45+63+93+105+155+217+279+315+465+651+1085+1395+1953+3255 24: 10395 -> 12645 = 1+3+5+7+9+11+15+21+27+33+35+45+55+63+77+99+105+135+165+189+231+297+315+385+495+693+945+1155+1485+2079+3465 25: 11025 -> 11946 = 1+3+5+7+9+15+21+25+35+45+49+63+75+105+147+175+225+245+315+441+525+735+1225+1575+2205+3675 The one thousandth abundant odd number is: 492975 and the proper divisor sum is: 519361 The first abundant odd number above one billion is: 1000000575 and the proper divisor sum is: 1083561009
Python
Procedural
<lang Python>#!/usr/bin/python
- Abundant odd numbers - Python
oddNumber = 1 aCount = 0 dSum = 0
from math import sqrt
def divisorSum(n):
sum = 1
i = int(sqrt(n)+1)
for d in range (2, i):
if n % d == 0:
sum += d
otherD = n // d
if otherD != d:
sum += otherD
return sum
print ("The first 25 abundant odd numbers:") while aCount < 25:
dSum = divisorSum(oddNumber )
if dSum > oddNumber :
aCount += 1
print("{0:5} proper divisor sum: {1}". format(oddNumber ,dSum ))
oddNumber += 2
while aCount < 1000:
dSum = divisorSum(oddNumber )
if dSum > oddNumber :
aCount += 1
oddNumber += 2
print ("\n1000th abundant odd number:") print (" ",(oddNumber - 2)," proper divisor sum: ",dSum)
oddNumber = 1000000001 found = False while not found :
dSum = divisorSum(oddNumber )
if dSum > oddNumber :
found = True
print ("\nFirst abundant odd number > 1 000 000 000:")
print (" ",oddNumber," proper divisor sum: ",dSum)
oddNumber += 2</lang>
- Output:
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4513
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 5977
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
Functional
<lang python>Odd abundant numbers
from math import sqrt from itertools import chain, count, islice
- abundantTuple :: Int -> [(Int, Int)]
def abundantTuple(n):
A list containing the tuple of N and its divisor
sum, if n is abundant, or an empty list.
x = divisorSum(n)
return [(n, x)] if n < x else []
- divisorSum :: Int -> [Int]
def divisorSum(n):
Sum of the divisors of n.
floatRoot = sqrt(n)
intRoot = int(floatRoot)
blnSquare = intRoot == floatRoot
lows = [x for x in range(1, 1 + intRoot) if 0 == n % x]
return sum(lows + [
n // x for x in (
lows[1:-1] if blnSquare else lows[1:]
)
])
- TEST ----------------------------------------------------
- main :: IO ()
def main():
Subsets of abundant odd numbers.
# First 25.
print('First 25 abundant odd numbers with their divisor sums:')
for x in take(25)(
concatMap(abundantTuple)(
enumFromThen(1)(3)
)
):
print(x)
# The 1000th.
print('\n1000th odd abundant number with its divisor sum:')
print(
take(1000)(
concatMap(abundantTuple)(
enumFromThen(1)(3)
)
)[-1]
)
# First over 10^9.
print('\nFirst odd abundant number over 10^9, with its divisor sum:')
billion = (10 ** 9)
print(
take(1)(
concatMap(abundantTuple)(
enumFromThen(1 + billion)(3 + billion)
)
)[0]
)
- GENERAL FUNCTIONS ---------------------------------------
- enumFromThen :: Int -> Int -> [Int]
def enumFromThen(m):
A non-finite stream of integers
starting at m, and continuing
at the interval between m and n.
return lambda n: count(m, n - m)
- concatMap :: (a -> [b]) -> [a] -> [b]
def concatMap(f):
A concatenated list over which a function f
has been mapped.
The list monad can be derived by using an (a -> [b])
function which wraps its output in a list (using an
empty list to represent computational failure).
return lambda xs: (
chain.from_iterable(map(f, xs))
)
- take :: Int -> [a] -> [a]
def take(n):
The prefix of xs of length n,
or xs itself if n > length xs.
return lambda xs: (
list(islice(xs, n))
)
if __name__ == '__main__':
main()</lang>
- Output:
First 25 abundant odd numbers with their divisor sums: (945, 975) (1575, 1649) (2205, 2241) (2835, 2973) (3465, 4023) (4095, 4641) (4725, 5195) (5355, 5877) (5775, 6129) (5985, 6495) (6435, 6669) (6615, 7065) (6825, 7063) (7245, 7731) (7425, 7455) (7875, 8349) (8085, 8331) (8415, 8433) (8505, 8967) (8925, 8931) (9135, 9585) (9555, 9597) (9765, 10203) (10395, 12645) (11025, 11946) 1000th odd abundant number with its divisor sum: (492975, 519361) First odd abundant number over 10^9, with its divisor sum: (1000000575, 1083561009)
Racket
<lang racket>#lang racket
(require math/number-theory
racket/generator)
(define (make-generator start)
(in-generator
(for ([n (in-naturals start)] #:when (odd? n))
(define divisor-sum (- (apply + (divisors n)) n))
(when (> divisor-sum n) (yield (list n divisor-sum))))))
(for/list ([i (in-range 25)] [x (make-generator 0)]) x) ; Task 1 (for/last ([i (in-range 1000)] [x (make-generator 0)]) x) ; Task 2 (for/first ([x (make-generator (add1 (inexact->exact 1e9)))]) x) ; Task 3</lang>
- Output:
'((945 975) (1575 1649) (2205 2241) (2835 2973) (3465 4023) (4095 4641) (4725 5195) (5355 5877) (5775 6129) (5985 6495) (6435 6669) (6615 7065) (6825 7063) (7245 7731) (7425 7455) (7875 8349) (8085 8331) (8415 8433) (8505 8967) (8925 8931) (9135 9585) (9555 9597) (9765 10203) (10395 12645) (11025 11946)) '(492975 519361) '(1000000575 1083561009)
REXX
A wee bit of coding was added to add commas (because of the larger numbers) as well as alignment of the output. <lang rexx>/*REXX pgm displays abundant odd numbers: 1st 25, one─thousandth, first > 1 billion. */ parse arg Nlow Nuno Novr . /*obtain optional arguments from the CL*/ if Nlow== | Nlow=="," then Nlow= 25 /*Not specified? Then use the default.*/ if Nuno== | Nuno=="," then Nuno= 1000 /* " " " " " " */ if Novr== | Novr=="," then Novr= 1000000000 /* " " " " " " */ numeric digits max(9, length(Novr) ) /*ensure enough decimal digits for // */ @= 'odd abundant number' /*variable for annotating the output. */
- = 0 /*count of odd abundant numbers so far.*/
do j=3 by 2 until #>=Nlow; $= sigO(j) /*get the sigma for an odd integer. */
if $<=j then iterate /*sigma ≤ J ? Then ignore it. */
#= # + 1 /*bump the counter for abundant odd #'s*/
say rt(th(#)) @ 'is:'rt(commas(j), 8) rt('sigma=') rt(commas($), 9)
end /*j*/
say
- = 0 /*count of odd abundant numbers so far.*/
do j=3 by 2; $= sigO(j) /*get the sigma for an odd integer. */
if $<=j then iterate /*sigma ≤ J ? Then ignore it. */
#= # + 1 /*bump the counter for abundant odd #'s*/
if #<Nuno then iterate /*Odd abundant# count<Nuno? Then skip.*/
say rt(th(#)) @ 'is:'rt(commas(j), 8) rt('sigma=') rt(commas($), 9)
leave /*we're finished displaying NUNOth num.*/
end /*j*/
say
do j=1+Novr%2*2 by 2; $= sigO(j) /*get sigma for an odd integer > Novr. */
if $<=j then iterate /*sigma ≤ J ? Then ignore it. */
say rt(th(1)) @ 'over' commas(Novr) "is: " commas(j) rt('sigma=') commas($)
leave /*we're finished displaying NOVRth num.*/
end /*j*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas:parse arg _; do c_=length(_)-3 to 1 by -3; _=insert(',', _, c_); end; return _ rt: procedure; parse arg #,len; if len== then len= 20; return right(#, len) th: parse arg th; return th||word('th st nd rd',1+(th//10)*(th//100%10\==1)*(th//10<4)) /*──────────────────────────────────────────────────────────────────────────────────────*/ sigO: procedure; parse arg x; s= 1 /*sigma for odd integers. ___*/
do k=3 by 2 while k*k<x /*divide by all odd integers up to √ x */
if x//k==0 then s= s + k + x%k /*add the two divisors to (sigma) sum. */
end /*k*/ /* ___*/
if k*k==x then return s + k /*Was X a square? If so, add √ x */
return s /*return (sigma) sum of the divisors. */</lang>
- output when using the default input:
1st odd abundant number is: 945 sigma= 975
2nd odd abundant number is: 1,575 sigma= 1,649
3rd odd abundant number is: 2,205 sigma= 2,241
4th odd abundant number is: 2,835 sigma= 2,973
5th odd abundant number is: 3,465 sigma= 4,023
6th odd abundant number is: 4,095 sigma= 4,641
7th odd abundant number is: 4,725 sigma= 5,195
8th odd abundant number is: 5,355 sigma= 5,877
9th odd abundant number is: 5,775 sigma= 6,129
10th odd abundant number is: 5,985 sigma= 6,495
11th odd abundant number is: 6,435 sigma= 6,669
12th odd abundant number is: 6,615 sigma= 7,065
13th odd abundant number is: 6,825 sigma= 7,063
14th odd abundant number is: 7,245 sigma= 7,731
15th odd abundant number is: 7,425 sigma= 7,455
16th odd abundant number is: 7,875 sigma= 8,349
17th odd abundant number is: 8,085 sigma= 8,331
18th odd abundant number is: 8,415 sigma= 8,433
19th odd abundant number is: 8,505 sigma= 8,967
20th odd abundant number is: 8,925 sigma= 8,931
21st odd abundant number is: 9,135 sigma= 9,585
22nd odd abundant number is: 9,555 sigma= 9,597
23rd odd abundant number is: 9,765 sigma= 10,203
24th odd abundant number is: 10,395 sigma= 12,645
25th odd abundant number is: 11,025 sigma= 11,946
1000th odd abundant number is: 492,975 sigma= 519,361
1st odd abundant number over 1,000,000,000 is: 1,000,000,575 sigma= 1,083,561,009
Ring
<lang ring>
- Project: Anbundant odd numbers
max = 100000000 limit = 25 nr = 0 m = 1 check = 0 index = 0 see "working..." + nl see "wait for done..." + nl while true
check = 0
if m%2 = 1
nice(m)
ok
if check = 1
nr = nr + 1
ok
if nr = max
exit
ok
m = m + 1
end see "done..." + nl
func nice(n)
check = 0
nArray = []
for i = 1 to n - 1
if n % i = 0
add(nArray,i)
ok
next
sum = 0
for p = 1 to len(nArray)
sum = sum + nArray[p]
next
if sum > n
check = 1
index = index + 1
if index < limit + 1
showArray(n,nArray,sum,index)
ok
if index = 100
see "One thousandth abundant odd number:" + nl
showArray2(n,nArray,sum,index)
ok
if index = 100000000
see "First abundant odd number above one billion:" + nl
showArray2(n,nArray,sum,index)
ok
ok
func showArray(n,nArray,sum,index)
see "" + index + ". " + string(n) + ": divisor sum: "
for m = 1 to len(nArray)
if m < len(nArray)
see string(nArray[m]) + " + "
else
see string(nArray[m]) + " = " + string(sum) + nl + nl
ok
next
func showArray2(n,nArray,sum,index)
see "" + index + ". " + string(n) + ": divisor sum: " +
see string(nArray[m]) + " = " + string(sum) + nl + nl
</lang>
working... wait for done... 1. 945: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095: divisor sum: 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435: divisor sum: 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085: divisor sum: 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415: divisor sum: 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925: divisor sum: 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555: divisor sum: 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395: divisor sum: 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 One thousandth abundant odd number: 1000. 492975: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 First abundant odd number above one billion: 100000000. 1000000575: divisor sum: 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009 done...
Ruby
proper_divisors method taken from http://rosettacode.org/wiki/Proper_divisors#Ruby <lang ruby>require "prime"
class Integer
def proper_divisors
return [] if self == 1
primes = prime_division.flat_map{|prime, freq| [prime] * freq}
(1...primes.size).each_with_object([1]) do |n, res|
primes.combination(n).map{|combi| res << combi.inject(:*)}
end.flatten.uniq
end
end
def generator_odd_abundants(from=1)
from += 1 if from.even?
Enumerator.new do |y|
from.step(nil, 2) do |n|
sum = n.proper_divisors.sum
y << [n, sum] if sum > n
end
end
end
generator_odd_abundants.take(25).each{|n, sum| puts "#{n} with sum #{sum}" } puts "\n%d with sum %#d" % generator_odd_abundants.take(1000).last puts "\n%d with sum %#d" % generator_odd_abundants(1_000_000_000).next </lang>
Rust
<lang rust>fn divisors(n: u64) -> Vec<u64> {
let mut divs = vec![1]; let mut divs2 = Vec::new();
for i in (2..).take_while(|x| x * x <= n).filter(|x| n % x == 0) {
divs.push(i);
let j = n / i;
if i != j {
divs2.push(j);
}
}
divs.extend(divs2.iter().rev());
divs
}
fn sum_string(v: Vec<u64>) -> String {
v[1..]
.iter()
.fold(format!("{}", v[0]), |s, i| format!("{} + {}", s, i))
}
fn abundant_odd(search_from: u64, count_from: u64, count_to: u64, print_one: bool) -> u64 {
let mut count = count_from;
for n in (search_from..).step_by(2) {
let divs = divisors(n);
let total: u64 = divs.iter().sum();
if total > n {
count += 1;
let s = sum_string(divs);
if !print_one {
println!("{}. {} < {} = {}", count, n, s, total);
} else if count == count_to {
println!("{} < {} = {}", n, s, total);
}
}
if count == count_to {
break;
}
}
count_to
}
fn main() {
let max = 25;
println!("The first {} abundant odd numbers are:", max);
let n = abundant_odd(1, 0, max, false);
println!("The one thousandth abundant odd number is:");
abundant_odd(n, 25, 1000, true);
println!("The first abundant odd number above one billion is:");
abundant_odd(1e9 as u64 + 1, 0, 1, true);
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 479115 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 27 + 35 + 39 + 45 + 63 + 65 + 81 + 91 + 105 + 117 + 135 + 169 + 189 + 195 + 273 + 315 + 351 + 405 + 455 + 507 + 567 + 585 + 819 + 845 + 945 + 1053 + 1183 + 1365 + 1521 + 1755 + 2457 + 2535 + 2835 + 3549 + 4095 + 4563 + 5265 + 5915 + 7371 + 7605 + 10647 + 12285 + 13689 + 17745 + 22815 + 31941 + 36855 + 53235 + 68445 + 95823 + 159705 = 583749 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Scala
<lang scala>import scala.collection.mutable.ListBuffer
object Abundant {
def divisors(n: Int): ListBuffer[Int] = {
val divs = new ListBuffer[Int]
divs.append(1)
val divs2 = new ListBuffer[Int] var i = 2
while (i * i <= n) {
if (n % i == 0) {
val j = n / i
divs.append(i)
if (i != j) {
divs2.append(j)
}
}
i += 1
}
divs.appendAll(divs2.reverse) divs }
def abundantOdd(searchFrom: Int, countFrom: Int, countTo: Int, printOne: Boolean): Int = {
var count = countFrom
var n = searchFrom
while (count < countTo) {
val divs = divisors(n)
val tot = divs.sum
if (tot > n) {
count += 1
if (!printOne || !(count < countTo)) {
val s = divs.map(a => a.toString).mkString(" + ")
if (printOne) {
printf("%d < %s = %d\n", n, s, tot)
} else {
printf("%2d. %5d < %s = %d\n", count, n, s, tot)
}
}
}
n += 2
}
n }
def main(args: Array[String]): Unit = {
val max = 25
printf("The first %d abundant odd numbers are:\n", max)
val n = abundantOdd(1, 0, max, printOne = false)
printf("\nThe one thousandth abundant odd number is:\n")
abundantOdd(n, 25, 1000, printOne = true)
printf("\nThe first abundant odd number above one billion is:\n")
abundantOdd((1e9 + 1).intValue(), 0, 1, printOne = true)
}
}</lang>
- Output:
The first 25 abundant odd numbers are: 1. 945 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 = 975 2. 1575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 = 1649 3. 2205 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 = 2241 4. 2835 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 = 2973 5. 3465 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 = 4023 6. 4095 < 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 = 4641 7. 4725 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 = 5195 8. 5355 < 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 = 5877 9. 5775 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 = 6129 10. 5985 < 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 = 6495 11. 6435 < 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 = 6669 12. 6615 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 = 7065 13. 6825 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 = 7063 14. 7245 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 = 7731 15. 7425 < 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 = 7455 16. 7875 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 = 8349 17. 8085 < 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 = 8331 18. 8415 < 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 = 8433 19. 8505 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 = 8967 20. 8925 < 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 = 8931 21. 9135 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 = 9585 22. 9555 < 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 = 9597 23. 9765 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 = 10203 24. 10395 < 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 = 12645 25. 11025 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 = 11946 The one thousandth abundant odd number is: 492975 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 = 519361 The first abundant odd number above one billion is: 1000000575 < 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525 = 1083561009
Sidef
<lang ruby>func is_abundant(n) {
n.sigma > 2*n
}
func odd_abundants (from = 1) {
from = (from + 2)//3
from += (from%2 - 1)
3*from .. Inf `by` 6 -> lazy.grep(is_abundant)
}
say " Index | Number | proper divisor sum" const sep = "-------+-------------+-------------------\n" const fstr = "%6s | %11s | %11s\n"
print sep
odd_abundants().first(25).each_kv {|k,n|
printf(fstr, k+1, n, n.sigma-n)
}
with (odd_abundants().nth(1000)) {|n|
printf(sep + fstr, 1000, n, n.sigma-n)
}
with(odd_abundants(1e9).first) {|n|
printf(sep + fstr, '***', n, n.sigma-n)
}</lang>
- Output:
Index | Number | proper divisor sum
-------+-------------+-------------------
1 | 945 | 975
2 | 1575 | 1649
3 | 2205 | 2241
4 | 2835 | 2973
5 | 3465 | 4023
6 | 4095 | 4641
7 | 4725 | 5195
8 | 5355 | 5877
9 | 5775 | 6129
10 | 5985 | 6495
11 | 6435 | 6669
12 | 6615 | 7065
13 | 6825 | 7063
14 | 7245 | 7731
15 | 7425 | 7455
16 | 7875 | 8349
17 | 8085 | 8331
18 | 8415 | 8433
19 | 8505 | 8967
20 | 8925 | 8931
21 | 9135 | 9585
22 | 9555 | 9597
23 | 9765 | 10203
24 | 10395 | 12645
25 | 11025 | 11946
-------+-------------+-------------------
1000 | 492975 | 519361
-------+-------------+-------------------
*** | 1000000575 | 1083561009
Visual Basic .NET
<lang vbnet>Module AbundantOddNumbers
' find some abundant odd numbers - numbers where the sum of the proper ' divisors is bigger than the number ' itself
' returns the sum of the proper divisors of n
Private Function divisorSum(n As Integer) As Integer
Dim sum As Integer = 1
For d As Integer = 2 To Math.Round(Math.Sqrt(n))
If n Mod d = 0 Then
sum += d
Dim otherD As Integer = n \ d
IF otherD <> d Then
sum += otherD
End If
End If
Next d
Return sum
End Function
' find numbers required by the task
Public Sub Main(args() As String)
' first 25 odd abundant numbers
Dim oddNumber As Integer = 1
Dim aCount As Integer = 0
Dim dSum As Integer = 0
Console.Out.WriteLine("The first 25 abundant odd numbers:")
Do While aCount < 25
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
aCount += 1
Console.Out.WriteLine(oddNumber.ToString.PadLeft(6) & " proper divisor sum: " & dSum)
End If
oddNumber += 2
Loop
' 1000th odd abundant number
Do While aCount < 1000
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
aCount += 1
End If
oddNumber += 2
Loop
Console.Out.WriteLine("1000th abundant odd number:")
Console.Out.WriteLine(" " & (oddNumber - 2) & " proper divisor sum: " & dSum)
' first odd abundant number > one billion
oddNumber = 1000000001
Dim found As Boolean = False
Do While Not found
dSum = divisorSum(oddNumber)
If dSum > oddNumber Then
found = True
Console.Out.WriteLine("First abundant odd number > 1 000 000 000:")
Console.Out.WriteLine(" " & oddNumber & " proper divisor sum: " & dSum)
End If
oddNumber += 2
Loop
End Sub
End Module</lang>
- Output:
The first 25 abundant odd numbers:
945 proper divisor sum: 975
1575 proper divisor sum: 1649
2205 proper divisor sum: 2241
2835 proper divisor sum: 2973
3465 proper divisor sum: 4023
4095 proper divisor sum: 4641
4725 proper divisor sum: 5195
5355 proper divisor sum: 5877
5775 proper divisor sum: 6129
5985 proper divisor sum: 6495
6435 proper divisor sum: 6669
6615 proper divisor sum: 7065
6825 proper divisor sum: 7063
7245 proper divisor sum: 7731
7425 proper divisor sum: 7455
7875 proper divisor sum: 8349
8085 proper divisor sum: 8331
8415 proper divisor sum: 8433
8505 proper divisor sum: 8967
8925 proper divisor sum: 8931
9135 proper divisor sum: 9585
9555 proper divisor sum: 9597
9765 proper divisor sum: 10203
10395 proper divisor sum: 12645
11025 proper divisor sum: 11946
1000th abundant odd number:
492975 proper divisor sum: 519361
First abundant odd number > 1 000 000 000:
1000000575 proper divisor sum: 1083561009
zkl
<lang zkl>fcn oddAbundants(startAt=3){ //--> iterator
Walker.zero().tweak(fcn(rn){
n:=rn.value;
while(True){
sum:=0; foreach d in ([3.. n.toFloat().sqrt().toInt(), 2]){ if( (y:=n/d) *d != n) continue; sum += ((y==d) and y or y+d) } if(sum>n){ rn.set(n+2); return(n) } n+=2;
} }.fp(Ref(startAt.isOdd and startAt or startAt+1)))
}</lang> <lang zkl>fcn oddDivisors(n){ // -->sorted List
[3.. n.toFloat().sqrt().toInt(), 2].pump(List(1),'wrap(d){
if( (y:=n/d) *d != n) return(Void.Skip);
if (y==d) y else T(y,d)
}).flatten().sort()
} fcn printOAs(oas){ // List | int
foreach n in (vm.arglist.flatten()){
ds:=oddDivisors(n);
println("%6,d: %6,d = %s".fmt(n, ds.sum(0), ds.sort().concat(" + ")))
}
}</lang> <lang zkl>oaw:=oddAbundants();
println("First 25 abundant odd numbers:"); oaw.walk(25) : printOAs(_);
println("\nThe one thousandth abundant odd number is:"); oaw.drop(1_000 - 25).value : printOAs(_);
println("\nThe first abundant odd number above one billion is:"); printOAs(oddAbundants(1_000_000_000).next());</lang>
- Output:
945: 975 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 105 + 135 + 189 + 315 1,575: 1,649 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 315 + 525 2,205: 2,241 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 35 + 45 + 49 + 63 + 105 + 147 + 245 + 315 + 441 + 735 2,835: 2,973 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 315 + 405 + 567 + 945 3,465: 4,023 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 165 + 231 + 315 + 385 + 495 + 693 + 1155 4,095: 4,641 = 1 + 3 + 5 + 7 + 9 + 13 + 15 + 21 + 35 + 39 + 45 + 63 + 65 + 91 + 105 + 117 + 195 + 273 + 315 + 455 + 585 + 819 + 1365 4,725: 5,195 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 27 + 35 + 45 + 63 + 75 + 105 + 135 + 175 + 189 + 225 + 315 + 525 + 675 + 945 + 1575 5,355: 5,877 = 1 + 3 + 5 + 7 + 9 + 15 + 17 + 21 + 35 + 45 + 51 + 63 + 85 + 105 + 119 + 153 + 255 + 315 + 357 + 595 + 765 + 1071 + 1785 5,775: 6,129 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 25 + 33 + 35 + 55 + 75 + 77 + 105 + 165 + 175 + 231 + 275 + 385 + 525 + 825 + 1155 + 1925 5,985: 6,495 = 1 + 3 + 5 + 7 + 9 + 15 + 19 + 21 + 35 + 45 + 57 + 63 + 95 + 105 + 133 + 171 + 285 + 315 + 399 + 665 + 855 + 1197 + 1995 6,435: 6,669 = 1 + 3 + 5 + 9 + 11 + 13 + 15 + 33 + 39 + 45 + 55 + 65 + 99 + 117 + 143 + 165 + 195 + 429 + 495 + 585 + 715 + 1287 + 2145 6,615: 7,065 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 49 + 63 + 105 + 135 + 147 + 189 + 245 + 315 + 441 + 735 + 945 + 1323 + 2205 6,825: 7,063 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 25 + 35 + 39 + 65 + 75 + 91 + 105 + 175 + 195 + 273 + 325 + 455 + 525 + 975 + 1365 + 2275 7,245: 7,731 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 23 + 35 + 45 + 63 + 69 + 105 + 115 + 161 + 207 + 315 + 345 + 483 + 805 + 1035 + 1449 + 2415 7,425: 7,455 = 1 + 3 + 5 + 9 + 11 + 15 + 25 + 27 + 33 + 45 + 55 + 75 + 99 + 135 + 165 + 225 + 275 + 297 + 495 + 675 + 825 + 1485 + 2475 7,875: 8,349 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 125 + 175 + 225 + 315 + 375 + 525 + 875 + 1125 + 1575 + 2625 8,085: 8,331 = 1 + 3 + 5 + 7 + 11 + 15 + 21 + 33 + 35 + 49 + 55 + 77 + 105 + 147 + 165 + 231 + 245 + 385 + 539 + 735 + 1155 + 1617 + 2695 8,415: 8,433 = 1 + 3 + 5 + 9 + 11 + 15 + 17 + 33 + 45 + 51 + 55 + 85 + 99 + 153 + 165 + 187 + 255 + 495 + 561 + 765 + 935 + 1683 + 2805 8,505: 8,967 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 27 + 35 + 45 + 63 + 81 + 105 + 135 + 189 + 243 + 315 + 405 + 567 + 945 + 1215 + 1701 + 2835 8,925: 8,931 = 1 + 3 + 5 + 7 + 15 + 17 + 21 + 25 + 35 + 51 + 75 + 85 + 105 + 119 + 175 + 255 + 357 + 425 + 525 + 595 + 1275 + 1785 + 2975 9,135: 9,585 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 29 + 35 + 45 + 63 + 87 + 105 + 145 + 203 + 261 + 315 + 435 + 609 + 1015 + 1305 + 1827 + 3045 9,555: 9,597 = 1 + 3 + 5 + 7 + 13 + 15 + 21 + 35 + 39 + 49 + 65 + 91 + 105 + 147 + 195 + 245 + 273 + 455 + 637 + 735 + 1365 + 1911 + 3185 9,765: 10,203 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 31 + 35 + 45 + 63 + 93 + 105 + 155 + 217 + 279 + 315 + 465 + 651 + 1085 + 1395 + 1953 + 3255 10,395: 12,645 = 1 + 3 + 5 + 7 + 9 + 11 + 15 + 21 + 27 + 33 + 35 + 45 + 55 + 63 + 77 + 99 + 105 + 135 + 165 + 189 + 231 + 297 + 315 + 385 + 495 + 693 + 945 + 1155 + 1485 + 2079 + 3465 11,025: 11,946 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 The one thousandth abundant odd number is: 492,975: 519,361 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 63 + 75 + 105 + 175 + 225 + 313 + 315 + 525 + 939 + 1565 + 1575 + 2191 + 2817 + 4695 + 6573 + 7825 + 10955 + 14085 + 19719 + 23475 + 32865 + 54775 + 70425 + 98595 + 164325 The first abundant odd number above one billion is: 1,000,000,575: 1,083,561,009 = 1 + 3 + 5 + 7 + 9 + 15 + 21 + 25 + 35 + 45 + 49 + 63 + 75 + 105 + 147 + 175 + 225 + 245 + 315 + 441 + 525 + 735 + 1225 + 1575 + 2205 + 3675 + 11025 + 90703 + 272109 + 453515 + 634921 + 816327 + 1360545 + 1904763 + 2267575 + 3174605 + 4081635 + 4444447 + 5714289 + 6802725 + 9523815 + 13333341 + 15873025 + 20408175 + 22222235 + 28571445 + 40000023 + 47619075 + 66666705 + 111111175 + 142857225 + 200000115 + 333333525