Arithmetic derivative
You are encouraged to solve this task according to the task description, using any language you may know.
The arithmetic derivative of an integer (more specifically, the Lagarias arithmetic derivative) is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. Accordingly, for natural numbers n, the arithmetic derivative D(n) is defined as follows:
- D(0) = D(1) = 0.
- D(p) = 1 for any prime p.
- D(mn) = D(m)n + mD(n) for any m,n ∈ N. (Leibniz rule for derivatives).
Additionally, for negative integers the arithmetic derivative may be defined as -D(-n) (n < 0).
- Examples
D(2) = 1 and D(3) = 1 (both are prime) so if mn = 2 * 3, D(6) = (1)(3) + (1)(2) = 5.
D(9) = D(3)(3) + D(3)(3) = 6
D(27) = D(3)*9 + D(9)*3 = 9 + 18 = 27
D(30) = D(5)(6) + D(6)(5) = 6 + 5 * 5 = 31.
- Task
Find and show the arithmetic derivatives for -99 through 100.
- Stretch task
Find (the arithmetic derivative of 10^m) then divided by 7, where m is from 1 to 20.
- See also
ABC
HOW TO RETURN lagarias n:
SELECT:
n<0: RETURN -lagarias -n
n in {0;1}: RETURN 0
NO d IN {2..floor root n} HAS n mod d=0: RETURN 1
ELSE: RETURN ((n/d) * lagarias d) + (d * lagarias (n/d))
PUT 0 IN col
FOR n IN {-99..100}:
WRITE (lagarias n)>>6
PUT col+1 IN col
IF col mod 10 = 0: WRITE/
FOR m IN {1..20}:
WRITE "D(10^`m>>2`) = `(lagarias (10**m))/7`"/- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10^ 1) = 1
D(10^ 2) = 20
D(10^ 3) = 300
D(10^ 4) = 4000
D(10^ 5) = 50000
D(10^ 6) = 600000
D(10^ 7) = 7000000
D(10^ 8) = 80000000
D(10^ 9) = 900000000
D(10^10) = 10000000000
D(10^11) = 110000000000
D(10^12) = 1200000000000
D(10^13) = 13000000000000
D(10^14) = 140000000000000
D(10^15) = 1500000000000000
D(10^16) = 16000000000000000
D(10^17) = 170000000000000000
D(10^18) = 1800000000000000000
D(10^19) = 19000000000000000000
D(10^20) = 200000000000000000000
Action!
PROC Main()
INT n, f, l, z
FOR n = -99 TO 100 DO
l = 0 f = 3 IF n < 0 THEN z = - n ELSE z = n FI
WHILE z >= 2 DO
WHILE z MOD 2 = 0 DO l ==+ n / 2 z ==/ 2 OD
IF f <= z THEN
WHILE z MOD f = 0 DO l ==+ n / f z ==/ f OD
f ==+ 2
FI
OD
PrintF( "%8I", l )
IF ( n + 100 ) MOD 10 = 0 THEN PutE() FI
OD
RETURN- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
ALGOL 68
BEGIN PROC lagarias = (LONG INT n) LONG INT: # Lagarias arithmetic derivative #
IF n < 0
THEN -lagarias (-n)
ELIF n = 0 OR n = 1
THEN 0
ELIF PROC small pf = (LONG INT j, k) LONG INT: # Smallest prime factor #
(j %* k = 0 | k | small pf (j, k + 1));
LONG INT f = small pf (n, 2); LONG INT q = n % f;
q = 1
THEN 1
ELSE q * lagarias (f) + f * lagarias (q)
FI;
FOR n FROM -99 TO 100
DO print (("D(", whole (n, 0), ") = ", whole (lagarias (n), 0), new line))
OD;
new line (standout);
FOR n TO 20
DO LONG INT m = LONG 10 ^ n;
print (("D(", whole (m, 0), ") / 7 = ", whole (lagarias (m) % 7, 0), new line))
OD
END- Output:
D(-99) = -75 D(-98) = -77 D(-97) = -1 D(-96) = -272 ... D(96) = 272 D(97) = 1 D(98) = 77 D(99) = 75 D(100) = 140 D(10) / 7 = 1 D(100) / 7 = 20 D(1000) / 7 = 300 ... D(1000000000000000000) / 7 = 1800000000000000000 D(10000000000000000000) / 7 = 19000000000000000000 D(100000000000000000000) / 7 = 200000000000000000000
Non-recursive alternative version
FOR n FROM -99 TO 100 DO
INT l := 0, f := 3, z := ABS n;
WHILE z >= 2 DO
WHILE z MOD 2 = 0 DO l +:= n OVER 2; z OVERAB 2 OD;
IF f <= z THEN
WHILE z MOD f = 0 DO l +:= n OVER f; z OVERAB f OD;
f +:= 2
FI
OD;
print( ( whole( l, -8 ) ) );
IF ( n + 100 ) MOD 10 = 0 THEN print( ( newline ) ) FI
OD- Output:
Same as the Action! sample.
ALGOL W
begin
integer procedure lagarias ( integer value n ) ; % Lagarias arithmetic derivative %
if n < 0
then -lagarias (-n)
else if n = 0 or n = 1
then 0
else begin
integer f, q;
integer procedure smallPf ( integer value j, k ) ; % Smallest prime factor %
if j rem k = 0 then k else smallPf (j, k + 1);
f := smallPf (n, 2); q := n div f;
if q = 1
then 1
else q * lagarias (f) + f * lagarias (q)
end lagarias ;
for n := -99 until 100 do begin
writeon( i_w := 6, s_w := 0, " ", lagarias (n) );
if n rem 10 = 0 then write()
end for_n
end.- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
APL
lagarias←{
⍵<0:-∇-⍵
⍵∊0 1:0
0=d←⊃1+⍸0=(1↓⍳⌊⍵*÷2)|⍵:1
(n×∇d)+d×∇n←⍵÷d
}
lagarias¨ 20 10⍴¯100+⍳200
- Output:
¯75 ¯77 ¯1 ¯272 ¯24 ¯49 ¯34 ¯96 ¯20 ¯123 ¯1 ¯140 ¯32 ¯45 ¯22 ¯124 ¯1 ¯43 ¯108 ¯176 ¯1 ¯71 ¯18 ¯80 ¯55 ¯39 ¯1 ¯156 ¯1 ¯59 ¯26 ¯72 ¯1 ¯61 ¯18 ¯192 ¯51 ¯33 ¯1 ¯92 ¯1 ¯31 ¯22 ¯92 ¯16 ¯81 ¯1 ¯56 ¯20 ¯45 ¯14 ¯112 ¯1 ¯25 ¯39 ¯48 ¯1 ¯41 ¯1 ¯68 ¯16 ¯21 ¯1 ¯60 ¯12 ¯19 ¯14 ¯80 ¯1 ¯31 ¯1 ¯32 ¯27 ¯15 ¯10 ¯44 ¯1 ¯13 ¯10 ¯24 ¯1 ¯21 ¯1 ¯32 ¯8 ¯9 ¯1 ¯16 ¯1 ¯7 ¯6 ¯12 ¯1 ¯5 ¯1 ¯4 ¯1 ¯1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140
BASIC
10 DEFINT A-Z
20 FOR N=-99 TO 100
30 GOSUB 100: PRINT USING "########";L;
40 NEXT
50 END
100 L=0: F=3: Z=ABS(N)
110 IF Z<2 THEN RETURN
120 IF Z MOD 2=0 THEN L=L+N\2: Z=Z\2: GOTO 120
130 IF F>Z THEN RETURN
140 IF Z MOD F=0 THEN L=L+N\F: Z=Z\F: GOTO 140
150 F=F+2
160 GOTO 130
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
C
#include <stdio.h>
#include <stdint.h>
typedef uint64_t u64;
void primeFactors(u64 n, u64 *factors, int *length) {
if (n < 2) return;
int count = 0;
int inc[8] = {4, 2, 4, 2, 4, 6, 2, 6};
while (!(n%2)) {
factors[count++] = 2;
n /= 2;
}
while (!(n%3)) {
factors[count++] = 3;
n /= 3;
}
while (!(n%5)) {
factors[count++] = 5;
n /= 5;
}
for (u64 k = 7, i = 0; k*k <= n; ) {
if (!(n%k)) {
factors[count++] = k;
n /= k;
} else {
k += inc[i];
i = (i + 1) % 8;
}
}
if (n > 1) {
factors[count++] = n;
}
*length = count;
}
double D(double n) {
if (n < 0) return -D(-n);
if (n < 2) return 0;
int i, length;
double d;
u64 f[80], g;
if (n < 1e19) {
primeFactors((u64)n, f, &length);
} else {
g = (u64)(n / 100);
primeFactors(g, f, &length);
f[length+1] = f[length] = 2;
f[length+3] = f[length+2] = 5;
length += 4;
}
if (length == 1) return 1;
if (length == 2) return (double)(f[0] + f[1]);
d = n / (double)f[0];
return D(d) * (double)f[0] + d;
}
int main() {
u64 ad[200];
int n, m;
double pow;
for (n = -99; n < 101; ++n) {
ad[n+99] = (int)D((double)n);
}
for (n = 0; n < 200; ++n) {
printf("%4ld ", ad[n]);
if (!((n+1)%10)) printf("\n");
}
printf("\n");
pow = 1;
for (m = 1; m < 21; ++m) {
pow *= 10;
printf("D(10^%-2d) / 7 = %.0f\n", m, D(pow)/7);
}
return 0;
}
- Output:
As Go example
C++
#include <iomanip>
#include <iostream>
#include <boost/multiprecision/cpp_int.hpp>
template <typename IntegerType>
IntegerType arithmetic_derivative(IntegerType n) {
bool negative = n < 0;
if (negative)
n = -n;
if (n < 2)
return 0;
IntegerType sum = 0, count = 0, m = n;
while ((m & 1) == 0) {
m >>= 1;
count += n;
}
if (count > 0)
sum += count / 2;
for (IntegerType p = 3, sq = 9; sq <= m; p += 2) {
count = 0;
while (m % p == 0) {
m /= p;
count += n;
}
if (count > 0)
sum += count / p;
sq += (p + 1) << 2;
}
if (m > 1)
sum += n / m;
if (negative)
sum = -sum;
return sum;
}
int main() {
using boost::multiprecision::int128_t;
for (int n = -99, i = 0; n <= 100; ++n, ++i) {
std::cout << std::setw(4) << arithmetic_derivative(n)
<< ((i + 1) % 10 == 0 ? '\n' : ' ');
}
int128_t p = 10;
std::cout << '\n';
for (int i = 0; i < 20; ++i, p *= 10) {
std::cout << "D(10^" << std::setw(2) << i + 1
<< ") / 7 = " << arithmetic_derivative(p) / 7 << '\n';
}
}
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^ 1) / 7 = 1 D(10^ 2) / 7 = 20 D(10^ 3) / 7 = 300 D(10^ 4) / 7 = 4000 D(10^ 5) / 7 = 50000 D(10^ 6) / 7 = 600000 D(10^ 7) / 7 = 7000000 D(10^ 8) / 7 = 80000000 D(10^ 9) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
CLU
factors = iter (n: bigint) yields (bigint)
own zero: bigint := bigint$i2bi(0)
own two: bigint := bigint$i2bi(2)
own three: bigint := bigint$i2bi(3)
while n>zero cand n//two = zero do yield(two) n := n/two end
fac: bigint := three
while fac<=n do
while n//fac = zero do yield(fac) n := n/fac end
fac := fac + two
end
end factors
lagarias = proc (n: bigint) returns (bigint)
own zero: bigint := bigint$i2bi(0)
if n < zero then return(-lagarias(-n)) end
sum: bigint := zero
for fac: bigint in factors(n) do
sum := sum + n / fac
end
return(sum)
end lagarias
start_up = proc ()
own po: stream := stream$primary_output()
own seven: bigint := bigint$i2bi(7)
own ten: bigint := bigint$i2bi(10)
for n: int in int$from_to(-99, 100) do
stream$putright(po, bigint$unparse(lagarias(bigint$i2bi(n))), 7)
if (n + 100)//10 = 0 then stream$putl(po, "") end
end
for m: int in int$from_to(1, 20) do
d: bigint := lagarias(ten ** bigint$i2bi(m)) / seven
stream$puts(po, "D(10^")
stream$putright(po, int$unparse(m), 2)
stream$puts(po, ") / 7 = ")
stream$putright(po, bigint$unparse(d), 25)
stream$putl(po, "")
end
end start_up- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10^ 1) / 7 = 1
D(10^ 2) / 7 = 20
D(10^ 3) / 7 = 300
D(10^ 4) / 7 = 4000
D(10^ 5) / 7 = 50000
D(10^ 6) / 7 = 600000
D(10^ 7) / 7 = 7000000
D(10^ 8) / 7 = 80000000
D(10^ 9) / 7 = 900000000
D(10^10) / 7 = 10000000000
D(10^11) / 7 = 110000000000
D(10^12) / 7 = 1200000000000
D(10^13) / 7 = 13000000000000
D(10^14) / 7 = 140000000000000
D(10^15) / 7 = 1500000000000000
D(10^16) / 7 = 16000000000000000
D(10^17) / 7 = 170000000000000000
D(10^18) / 7 = 1800000000000000000
D(10^19) / 7 = 19000000000000000000
D(10^20) / 7 = 200000000000000000000
Cowgol
include "cowgol.coh";
sub abs(n: int32): (r: uint32) is
if n<0 then
r := (-n) as uint32;
else
r := n as uint32;
end if;
end sub;
sub printcol(n: int32, s: uint8) is
var buf: uint8[12];
var ptr := IToA(n, 10, &buf[0]);
s := s - (ptr - &buf[0]) as uint8;
while s>0 loop
print_char(' ');
s := s-1;
end loop;
print(&buf[0]);
end sub;
sub lagarias(n: int32): (r: int32) is
var nn := abs(n);
r := 0;
if nn<2 then return; end if;
var f: uint32 := 2;
while f<=nn loop
while nn%f == 0 loop
r := r + n/f as int32;
nn := nn/f;
end loop;
f := f+1;
end loop;
end sub;
var i: int32 := -99;
var c: uint8 := 0;
while i <= 100 loop
printcol(lagarias(i), 7);
i := i+1;
c := c+1;
if c%10 == 0 then print_nl(); end if;
end loop;- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
Draco
proc lagarias(int n) int:
int f, r, s;
if n<0 then
-lagarias(-n)
elif n<2 then
0
else
s := 0;
r := n;
while r % 2 = 0 do
r := r / 2;
s := s + n / 2
od;
f := 3;
while f <= r do
while r % f = 0 do
r := r / f;
s := s + n / f
od;
f := f + 2
od;
s
fi
corp
proc main() void:
int n;
for n from -99 upto 100 do
write(lagarias(n):7);
if (n+100) % 10=0 then writeln() fi
od
corp- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
EasyLang
func lagarias n .
if n < 0
return -lagarias -n
.
if n = 0 or n = 1
return 0
.
f = 2
while n mod f <> 0
f += 1
.
q = n / f
if q = 1
return 1
.
return q * lagarias f + f * lagarias q
.
for n = -99 to 100
write lagarias n & " "
.
Factor
USING: combinators formatting grouping io kernel math
math.primes.factors prettyprint ranges sequences ;
: n' ( m -- n )
{
{ [ dup neg? ] [ neg n' neg ] }
{ [ dup 2 < ] [ drop 0 ] }
{ [ factors dup length 1 = ] [ drop 1 ] }
[ unclip-slice swap product 2dup n' * spin n' * + ]
} cond ;
-99 100 [a..b] [ n' ] map 10 group
[ [ "%5d" printf ] each nl ] each
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
FreeBASIC
Function aDerivative(Byval n As Longint) As Longint
If n < 0 Then Return -aDerivative(-n)
If n = 0 Or n = 1 Then Return 0
If n = 2 Then Return 1
Dim As Longint q, d = 2
Dim As Longint result = 1
While d * d <= n
If n Mod d = 0 Then
q = n \ d
result = q * aDerivative(d) + d * aDerivative(q)
Exit While
End If
d += 1
Wend
Return result
End Function
'Main program
Print "Arithmetic derivatives for -99 through 100:"
Dim As Integer col, n
col = 0
For n = -99 To 100
col += 1
Print Using "####"; aDerivative(n);
If col = 10 Then
Print
col = 0
Else
Print " ";
End If
Next
'Stretch task
Print !"\n\nPowers of 10 derivatives divided by 7:"
Dim As Double m = 1
For n = 1 To 18 ' LongInt limit in FreeBASIC
m *= 10
Dim As Longint a = aDerivative(Clngint(m))
Print Using "D(10^&) / 7 = &"; n; a \ 7
Next
Sleep
- Output:
Arithmetic derivatives for -99 through 100: -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 Powers of 10 derivatives divided by 7: D(10^1)/7 = 1 D(10^2)/7 = 20 D(10^3)/7 = 300 D(10^4)/7 = 4000 D(10^5)/7 = 50000 D(10^6)/7 = 600000 D(10^7)/7 = 7000000 D(10^8)/7 = 80000000 D(10^9)/7 = 900000000 D(10^10)/7 = 10000000000 D(10^11)/7 = 110000000000 D(10^12)/7 = 1200000000000 D(10^13)/7 = 13000000000000 D(10^14)/7 = 140000000000000 D(10^15)/7 = 1500000000000000 D(10^16)/7 = 16000000000000000 D(10^17)/7 = 170000000000000000
FutureBasic
// Arithmetic Derivative
// https://rosettacode.org/wiki/Arithmetic_derivative#
local fn DoIt( N as short) as short
short L,F,Z
L = 0: F = 3: Z = ABS(N)
IF Z<2 THEN exit fn
1 IF Z MOD 2 = 0 THEN L=L+N\2: Z=Z\2: GOTO 1
2 IF F>Z THEN exit fn
3 IF Z MOD F = 0 THEN L=L+N\F: Z=Z\F: GOTO 2
F=F+2
goto 1
end fn = L
_Window = 1
window _Window,@"Arithmetic Derivative",fn cgrectmake(0,0,640,400)
windowcenter(_Window)
short N,L,LineCount
FOR N = -99 TO 100
L = fn DoIt(N): PRINT USING "########";L;
LineCount ++
if LineCount = 10 then print : LineCount = 0
NEXT
handleevents- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140
Go
Using float64 (finessed a little) to avoid the unpleasantness of math/big for the stretch goal. Assumes that int type is 64 bit.
package main
import (
"fmt"
"rcu"
)
func D(n float64) float64 {
if n < 0 {
return -D(-n)
}
if n < 2 {
return 0
}
var f []int
if n < 1e19 {
f = rcu.PrimeFactors(int(n))
} else {
g := int(n / 100)
f = rcu.PrimeFactors(g)
f = append(f, []int{2, 2, 5, 5}...)
}
c := len(f)
if c == 1 {
return 1
}
if c == 2 {
return float64(f[0] + f[1])
}
d := n / float64(f[0])
return D(d)*float64(f[0]) + d
}
func main() {
ad := make([]int, 200)
for n := -99; n < 101; n++ {
ad[n+99] = int(D(float64(n)))
}
rcu.PrintTable(ad, 10, 4, false)
fmt.Println()
pow := 1.0
for m := 1; m < 21; m++ {
pow *= 10
fmt.Printf("D(10^%-2d) / 7 = %.0f\n", m, D(pow)/7)
}
}
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1 ) / 7 = 1 D(10^2 ) / 7 = 20 D(10^3 ) / 7 = 300 D(10^4 ) / 7 = 4000 D(10^5 ) / 7 = 50000 D(10^6 ) / 7 = 600000 D(10^7 ) / 7 = 7000000 D(10^8 ) / 7 = 80000000 D(10^9 ) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
Haskell
import Control.Monad (forM_)
import Data.List (intercalate)
import Data.List.Split (chunksOf)
import Math.NumberTheory.Primes (factorise, unPrime)
import Text.Printf (printf)
-- The arithmetic derivative of a number, which is assumed to be non-negative.
arithderiv_ :: Integer -> Integer
arithderiv_ 0 = 0
arithderiv_ n = foldr step 0 $ factorise n
where step (p, v) s = s + n `quot` unPrime p * fromIntegral v
-- The arithmetic derivative of any integer.
arithderiv :: Integer -> Integer
arithderiv n | n < 0 = negate $ arithderiv_ (negate n)
| otherwise = arithderiv_ n
printTable :: [Integer] -> IO ()
printTable = putStrLn
. intercalate "\n"
. map unwords
. chunksOf 10
. map (printf "%5d")
main :: IO ()
main = do
printTable [arithderiv n | n <- [-99..100]]
putStrLn ""
forM_ [1..20 :: Integer] $ \i ->
let q = 7
n = arithderiv (10^i) `quot` q
in printf "D(10^%-2d) / %d = %d\n" i q n
- Output:
$ arithderiv
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10^1 ) / 7 = 1
D(10^2 ) / 7 = 20
D(10^3 ) / 7 = 300
D(10^4 ) / 7 = 4000
D(10^5 ) / 7 = 50000
D(10^6 ) / 7 = 600000
D(10^7 ) / 7 = 7000000
D(10^8 ) / 7 = 80000000
D(10^9 ) / 7 = 900000000
D(10^10) / 7 = 10000000000
D(10^11) / 7 = 110000000000
D(10^12) / 7 = 1200000000000
D(10^13) / 7 = 13000000000000
D(10^14) / 7 = 140000000000000
D(10^15) / 7 = 1500000000000000
D(10^16) / 7 = 16000000000000000
D(10^17) / 7 = 170000000000000000
D(10^18) / 7 = 1800000000000000000
D(10^19) / 7 = 19000000000000000000
D(10^20) / 7 = 200000000000000000000
J
Implementation:
D=: {{ +/y%q:1>.|y }}"0
In other words: find the sum of the argument divided by each of the sequence of prime factors of its absolute value (with a special case for zero -- we use the maximum of either 1 or that absolute value when finding the sequence of prime factors).
Task example:
D _99+i.20 10
_75 _77 _1 _272 _24 _49 _34 _96 _20 _123
_1 _140 _32 _45 _22 _124 _1 _43 _108 _176
_1 _71 _18 _80 _55 _39 _1 _156 _1 _59
_26 _72 _1 _61 _18 _192 _51 _33 _1 _92
_1 _31 _22 _92 _16 _81 _1 _56 _20 _45
_14 _112 _1 _25 _39 _48 _1 _41 _1 _68
_16 _21 _1 _60 _12 _19 _14 _80 _1 _31
_1 _32 _27 _15 _10 _44 _1 _13 _10 _24
_1 _21 _1 _32 _8 _9 _1 _16 _1 _7
_6 _12 _1 _5 _1 _4 _1 _1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
Also, it seems like it's worth verifying that order of evaluation does not create an ambiguity for the value of D (order shouldn't matter, since summation of integers is order independent):
15 10 6 + 2 3 5 * D 15 10 6
31 31 31
Stretch task:
(D 10x^1+i.4 5)%7
1 20 300 4000 50000
600000 7000000 80000000 900000000 10000000000
110000000000 1200000000000 13000000000000 140000000000000 1500000000000000
16000000000000000 170000000000000000 1800000000000000000 19000000000000000000 200000000000000000000
Java
import java.math.BigInteger;
public final class ArithmeticDerivative {
public static void main(String[] aArgs) {
System.out.println("Arithmetic derivatives for -99 to 100 inclusive:");
for ( int n = -99, column = 0; n <= 100; n++ ) {
System.out.print(String.format("%4d%s",
derivative(BigInteger.valueOf(n)), ( ++column % 10 == 0 ) ? "\n" : " "));
}
System.out.println();
final BigInteger seven = BigInteger.valueOf(7);
for ( int power = 1; power <= 20; power++ ) {
System.out.println(String.format("%s%2d%s%d",
"D(10^", power, ") / 7 = ", derivative(BigInteger.TEN.pow(power)).divide(seven)));
}
}
private static BigInteger derivative(BigInteger aNumber) {
if ( aNumber.signum() == -1 ) {
return derivative(aNumber.negate()).negate();
}
if ( aNumber == BigInteger.ZERO || aNumber == BigInteger.ONE ) {
return BigInteger.ZERO;
}
BigInteger divisor = BigInteger.TWO;
while ( divisor.multiply(divisor).compareTo(aNumber) <= 0 ) {
if ( aNumber.mod(divisor).signum() == 0 ) {
final BigInteger quotient = aNumber.divide(divisor);
return quotient.multiply(derivative(divisor)).add(divisor.multiply(derivative(quotient)));
}
divisor = divisor.add(BigInteger.ONE);
}
return BigInteger.ONE;
}
}
- Output:
Arithmetic derivatives for -99 to 100 inclusive: -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 -1 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^ 1) / 7 = 1 D(10^ 2) / 7 = 20 D(10^ 3) / 7 = 300 D(10^ 4) / 7 = 4000 D(10^ 5) / 7 = 50000 D(10^ 6) / 7 = 600000 D(10^ 7) / 7 = 7000000 D(10^ 8) / 7 = 80000000 D(10^ 9) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
jq
For this task, gojq (the Go implementation of jq) is used for numerical accuracy, though the C implementation has sufficient accuracy at least for D(10^16).
See Prime_decomposition#jq for the def of factors/0 used here.
To take advantage of gojq's arbitrary-precision integer arithmetic:
def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);
# In case gojq is used:
def _nwise($n):
def nw: if length <= $n then . else .[0:$n] , (.[$n:] | nw) end;
nw;
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def D($n):
if $n < 0 then -D(- $n)
elif $n < 2 then 0
else [$n | factors] as $f
| ($f|length) as $c
| if $c <= 1 then 1
elif $c == 2 then $f[0] + $f[1]
else ($n / $f[0]) as $d
| D($d) * $f[0] + $d
end
end ;
def task:
def task1:
reduce range(-99; 101) as $n ([]; .[$n+99] = D($n))
| _nwise(10) | map(lpad(4)) | join(" ");
def task2:
range(1; 21) as $i
| "D(10^\($i)) / 7 = \( D(10|power($i))/7 )" ;
task1, "", task2 ;
task- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1) / 7 = 1 D(10^2) / 7 = 20 D(10^3) / 7 = 300 D(10^4) / 7 = 4000 D(10^5) / 7 = 50000 D(10^6) / 7 = 600000 D(10^7) / 7 = 7000000 D(10^8) / 7 = 80000000 D(10^9) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
Julia
using Primes
D(n) = n < 0 ? -D(-n) : n < 2 ? zero(n) : isprime(n) ? one(n) : typeof(n)(sum(e * n ÷ p for (p, e) in eachfactor(n)))
foreach(p -> print(lpad(p[2], 5), p[1] % 10 == 0 ? "\n" : ""), pairs(map(D, -99:100)))
println()
for m in 1:20
println("D for 10^", rpad(m, 3), "divided by 7 is ", D(Int128(10)^m) ÷ 7)
end
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D for 10^1 divided by 7 is 1
D for 10^2 divided by 7 is 20
D for 10^3 divided by 7 is 300
D for 10^4 divided by 7 is 4000
D for 10^5 divided by 7 is 50000
D for 10^6 divided by 7 is 600000
D for 10^7 divided by 7 is 7000000
D for 10^8 divided by 7 is 80000000
D for 10^9 divided by 7 is 900000000
D for 10^10 divided by 7 is 10000000000
D for 10^11 divided by 7 is 110000000000
D for 10^12 divided by 7 is 1200000000000
D for 10^13 divided by 7 is 13000000000000
D for 10^14 divided by 7 is 140000000000000
D for 10^15 divided by 7 is 1500000000000000
D for 10^16 divided by 7 is 16000000000000000
D for 10^17 divided by 7 is 170000000000000000
D for 10^18 divided by 7 is 1800000000000000000
D for 10^19 divided by 7 is 19000000000000000000
D for 10^20 divided by 7 is 200000000000000000000
Lua
Tested with Lua 5.1 (LuaJIT and OpenResty), 5.3.6 and 5.4.6.
Lua 5.2.4 didn't like the string.format. Also the format of the larger D values appears to be sensitive to the Lua version.
do local function lagarias (n) -- Lagarias arithmetic derivative
if n < 0
then return -lagarias (-n)
elseif n == 0 or n == 1
then return 0
else local function smallPf (j, k) -- Smallest prime factor
if j % k == 0 then return k else return smallPf (j, k + 1) end
end
local f = smallPf (n, 2) local q = math.floor (n / f)
if q == 1
then return 1
else return q * lagarias (f) + f * lagarias (q)
end
end
end
for n = -99,100
do io.write (string.format("%6d", lagarias (n)))
if n % 10 == 0 then io.write ("\n") end
end
io.write ("\n")
for n = 1,17 -- 18, 19 and 20 would overflow
do local m = 10 ^ n
io.write ("D(", string.format ("%d", m), ") / 7 = ", math.floor (lagarias (m) / 7), "\n")
end
end
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10) / 7 = 1
D(100) / 7 = 20
D(1000) / 7 = 300
D(10000) / 7 = 4000
D(100000) / 7 = 50000
D(1000000) / 7 = 600000
D(10000000) / 7 = 7000000
D(100000000) / 7 = 80000000
D(1000000000) / 7 = 900000000
D(10000000000) / 7 = 10000000000
D(100000000000) / 7 = 110000000000
D(1000000000000) / 7 = 1200000000000
D(10000000000000) / 7 = 13000000000000
D(100000000000000) / 7 = 140000000000000
D(1000000000000000) / 7 = 1500000000000000
D(10000000000000000) / 7 = 16000000000000000
D(100000000000000000) / 7 = 170000000000000000
MAD
NORMAL MODE IS INTEGER
INTERNAL FUNCTION(X,Y)
ENTRY TO REM.
FUNCTION RETURN X-(X/Y)*Y
END OF FUNCTION
INTERNAL FUNCTION(N)
ENTRY TO DERIV.
R = N
WHENEVER R.L.0, R = -R
WHENEVER R.L.2, FUNCTION RETURN 0
S = 0
FAC2 WHENEVER REM.(R,2).E.0
S = S + N/2
R = R/2
TRANSFER TO FAC2
END OF CONDITIONAL
THROUGH FAC, FOR F=3, 2, F.G.R
FACF WHENEVER REM.(R,F).E.0
S = S + N/F
R = R/F
TRANSFER TO FACF
END OF CONDITIONAL
FAC CONTINUE
FUNCTION RETURN S
END OF FUNCTION
VECTOR VALUES LINEF = $10(I6)*$
DIMENSION LINE(10)
C = 0
THROUGH ITEM, FOR I=-99, 1, I.G.100
LINE(C) = DERIV.(I)
C = C+1
WHENEVER C.E.10
PRINT FORMAT LINEF,
0 LINE(0),LINE(1),LINE(2),LINE(3),LINE(4),
1 LINE(5),LINE(6),LINE(7),LINE(8),LINE(9)
C = 0
END OF CONDITIONAL
ITEM CONTINUE
END OF PROGRAM- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
Mathematica / Wolfram Language
(* Arithmetic derivative *)
ClearAll[d, twoFactorsOf];
twoFactorsOf[n_Integer?Positive] := Module[{factors = FactorInteger[n, 2], p, factor},
If[Length[factors] == 1,
factor = Flatten@factors;
p = First@factor;
factors = {factor - {0, 1}, {p, 1}};
];
Return[Power@@@factors];
];
twoFactorsOf[n_Integer?Negative] := twoFactorsOf[-n] * {-1, -1};
d[0] = d[1] = 0;
d[p_Integer?PrimeQ] := 1;
d[n_Integer?Negative] := -d[-n];
d[mn_Integer] := Module[{m, n},
{m, n} = twoFactorsOf[m n];
Return[d[m] n + m d[n]];
];
SetAttributes[d, Listable];
(* Output *)
Partition[StringPadLeft[ToString /@ d[Range[-99, 100]], 5], UpTo[10]] // TableForm
StringJoin["d[10^", ToString@First[#], "]", If[First[#] <= 9, " ", " "], "/ 7",
" = ",
ToString@Last[#]] & /@ Table[{n, d[10^n]/7}, {n, 1, 20}] // TableForm
- Output:
57 -21 1 -80 -24 -49 -34 88 -20 -33
1 -52 -32 -45 -22 -68 1 -43 -54 64
1 59 -18 72 5 -39 1 12 1 39
-26 64 1 49 -18 -64 33 -33 1 -52
1 -31 -22 -36 -16 -45 1 48 -20 -5
-14 32 1 -25 21 40 1 29 1 -28
-16 -21 1 60 -12 -19 -14 16 1 19
1 24 9 -15 -10 -20 1 -13 -10 16
1 3 1 -16 -8 -9 1 8 1 -7
-6 4 1 -5 1 -4 1 1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
d[10^1] = 1
d[10^2] = 20
d[10^3] = 300
d[10^4] = 4000
d[10^5] = 50000
d[10^6] = 600000
d[10^7] = 7000000
d[10^8] = 80000000
d[10^9] = 900000000
d[10^10] = 10000000000
d[10^11] = 110000000000
d[10^12] = 1200000000000
d[10^13] = 13000000000000
d[10^14] = 140000000000000
d[10^15] = 1500000000000000
d[10^16] = 16000000000000000
d[10^17] = 170000000000000000
d[10^18] = 1800000000000000000
d[10^19] = 19000000000000000000
d[10^20] = 200000000000000000000
MiniScript
lagarias = function (n) // Lagarias arithmetic derivative
if n < 0 then
return -lagarias (-n)
else if n == 0 or n == 1 then
return 0
else
smallPf = function (j, k) // Smallest prime factor
if j % k == 0 then
return k
else
return smallPf (j, k + 1)
end if
end function
f = smallPf (n, 2)
q = floor (n / f)
if q == 1 then
return 1
else
return q * lagarias (f) + f * lagarias (q)
end if
end if
end function
fmt6 = function (n) // return a 6 character string representation of n
s = str( n )
if s.len > 5 then
return s
else
return ( " " * ( 6 - s.len ) ) + s
end if
end function
ad = ""
for n in range( -99, 100 )
ad = ad + " " + fmt6( lagarias (n) )
if n % 10 == 0 then
print( ad )
ad = ""
end if
end for
print()
for n in range( 1, 17 )
m = 10 ^ n
print( "D(" + str(m) + ") / 7 = " + str( floor (lagarias (m) / 7) ) )
end for
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10) / 7 = 1
D(100) / 7 = 20
D(1000) / 7 = 300
D(10000) / 7 = 4000
D(100000) / 7 = 50000
D(1000000) / 7 = 600000
D(10000000) / 7 = 7000000
D(100000000) / 7 = 80000000
D(1000000000) / 7 = 900000000
D(10000000000) / 7 = 10000000000
D(100000000000) / 7 = 110000000000
D(1000000000000) / 7 = 1200000000000
D(10000000000000) / 7 = 13000000000000
D(100000000000000) / 7 = 140000000000000
D(1000000000000000) / 7 = 1500000000000000
D(10000000000000000) / 7 = 16000000000000000
D(100000000000000000) / 7 = 170000000000000000
D(1000000000000000000) / 7 = 1800000000000000000
D(10000000000000000000) / 7 = 19000000000000000000
D(100000000000000000000) / 7 = 200000000000000000000
Miranda
main :: [sys_message]
main = [Stdout (table 10 7 (map (show . lagarias) [-99..100])),
Stdout (lay (map ten_pow_m_div_7 [1..20]))]
ten_pow_m_div_7 :: num->[char]
ten_pow_m_div_7 m = "D(10^" ++ rjustify 2 (show m) ++ ") / 7 = " ++
show (lagarias (10^m) div 7)
table :: num->num->[[char]]->[char]
table w cw ls = lay [concat (map (rjustify cw) l) | l <- group w ls]
group :: num->[*]->[[*]]
group n [] = []
group n ls = take n ls : group n (drop n ls)
lagarias :: num->num
lagarias n = -lagarias (-n), if n<0
= sum [n div f | f <- factors n], otherwise
factors :: num->[num]
factors n = f n 2
where f n d = [], if d > n
= d : f (n div d) d, if n mod d = 0
= f n (d+1), otherwise- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10^ 1) / 7 = 1
D(10^ 2) / 7 = 20
D(10^ 3) / 7 = 300
D(10^ 4) / 7 = 4000
D(10^ 5) / 7 = 50000
D(10^ 6) / 7 = 600000
D(10^ 7) / 7 = 7000000
D(10^ 8) / 7 = 80000000
D(10^ 9) / 7 = 900000000
D(10^10) / 7 = 10000000000
D(10^11) / 7 = 110000000000
D(10^12) / 7 = 1200000000000
D(10^13) / 7 = 13000000000000
D(10^14) / 7 = 140000000000000
D(10^15) / 7 = 1500000000000000
D(10^16) / 7 = 16000000000000000
D(10^17) / 7 = 170000000000000000
D(10^18) / 7 = 1800000000000000000
D(10^19) / 7 = 19000000000000000000
D(10^20) / 7 = 200000000000000000000
Nim
import std/[strformat, strutils]
import integers
func aDerivative(n: int | Integer): typeof(n) =
## Recursively compute the arithmetic derivative.
## The function works with normal integers or big integers.
## Using a cache to store the derivatives would improve the
## performance, but this is not needed for these tasks.
if n < 0: return -aDerivative(-n)
if n == 0 or n == 1: return 0
if n == 2: return 1
var d = 2
result = 1
while d * d <= n:
if n mod d == 0:
let q = n div d
result = q * aDerivative(d) + d * aDerivative(q)
break
inc d
### Task ###
echo "Arithmetic derivatives for -99 through 100:"
# We can use an "int" variable here.
var col = 0
for n in -99..100:
inc col
stdout.write &"{aDerivative(n):>4}"
stdout.write if col == 10: '\n' else: ' '
if col == 10: col = 0
### Stretch task ###
echo()
# To avoid overflow, we have to use an "Integer" variable.
var n = Integer(1)
for m in 1..20:
n *= 10
let a = aDerivative(n)
let left = &"D(10^{m}) / 7"
echo &"{left:>12} = {a div 7}"
- Output:
Arithmetic derivatives for -99 through 100: -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1) / 7 = 1 D(10^2) / 7 = 20 D(10^3) / 7 = 300 D(10^4) / 7 = 4000 D(10^5) / 7 = 50000 D(10^6) / 7 = 600000 D(10^7) / 7 = 7000000 D(10^8) / 7 = 80000000 D(10^9) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
Perl
use v5.36;
use bigint;
no warnings 'uninitialized';
use List::Util 'max';
use ntheory 'factor';
sub table ($c, @V) { my $t = $c * (my $w = 2 + length max @V); ( sprintf( ('%'.$w.'d')x@V, @V) ) =~ s/.{1,$t}\K/\n/gr }
sub D ($n) {
my(%f, $s);
$f{$_}++ for factor max 1, my $nabs = abs $n;
map { $s += $nabs * $f{$_} / $_ } keys %f;
$n > 0 ? $s : -$s;
}
say table 10, map { D $_ } -99 .. 100;
say join "\n", map { sprintf('D(10**%-2d) / 7 == ', $_) . D(10**$_) / 7 } 1 .. 20;
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10**1 ) / 7 == 1
D(10**2 ) / 7 == 20
D(10**3 ) / 7 == 300
D(10**4 ) / 7 == 4000
D(10**5 ) / 7 == 50000
D(10**6 ) / 7 == 600000
D(10**7 ) / 7 == 7000000
D(10**8 ) / 7 == 80000000
D(10**9 ) / 7 == 900000000
D(10**10) / 7 == 10000000000
D(10**11) / 7 == 110000000000
D(10**12) / 7 == 1200000000000
D(10**13) / 7 == 13000000000000
D(10**14) / 7 == 140000000000000
D(10**15) / 7 == 1500000000000000
D(10**16) / 7 == 16000000000000000
D(10**17) / 7 == 170000000000000000
D(10**18) / 7 == 1800000000000000000
D(10**19) / 7 == 19000000000000000000
D(10**20) / 7 == 200000000000000000000
PL/I
arithmeticDerivative: procedure options(main);
lagarias: procedure(n) returns(fixed);
declare (n, res, fac, rem) fixed;
rem = abs(n);
if rem<2 then return(0);
res = 0;
do while(mod(rem,2) = 0);
res = res + n/2;
rem = rem/2;
end;
do fac=3 repeat(fac+2) while(fac<=rem);
do while(mod(rem,fac) = 0);
res = res + n/fac;
rem = rem/fac;
end;
end;
return(res);
end lagarias;
declare n fixed;
do n=-99 to 100;
put edit(lagarias(n)) (F(7));
if mod(n+100, 10)=0 then put skip;
end;
end arithmeticDerivative;- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
PL/M
The 8080 PL/M only has unsigned integer arithmetic with integers in the range 0-65535. By adding 32768 we can compare values in the range -32768-32767 and get the correct result. The arithmetic operations +, - and * will produce the correct results but division and MOD need special handling if the operands are signed. MOD is not needed for this sample but the SIGNED$LT and SIGNED$DIV procedures implement signed < and / operations.
100H: /* ARITHMETIC DERIVATIVE - BASED ON THE BASIC SAMPLE */
/* RETURNS TRUE IF A < B, FALSE OTHERWISE WITH A AND B TREATED AS SIGNED */
SIGNED$LT: PROCEDURE( A, B )BYTE;
DECLARE ( A, B ) ADDRESS;
IF ( A + 32768 ) < ( B + 32768 ) THEN RETURN 0FFH; ELSE RETURN 0;
END SIGNED$LT ;
/* RETURNS A / B WITH A AND B TREATED AS SIGNED */
SIGNED$DIV: PROCEDURE( AIN, BIN )ADDRESS;
DECLARE ( AIN, BIN )ADDRESS;
DECLARE ( A, B, SIGN )ADDRESS;
SIGN = 1;
A = AIN;
B = BIN;
IF SIGNED$LT( A, 0 ) THEN DO;
SIGN = - SIGN;
A = - A;
END;
IF SIGNED$LT( B, 0 ) THEN DO;
SIGN = - SIGN;
B = - B;
END;
RETURN ( A / B ) * SIGN;
END SIGNED$DIV ;
/* CP/M BDOS SYSTEM CALL AND I/O ROUTINES */
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5; END;
PR$CHAR: PROCEDURE( C ); DECLARE C BYTE; CALL BDOS( 2, C ); END;
PR$STRING: PROCEDURE( S ); DECLARE S ADDRESS; CALL BDOS( 9, S ); END;
PR$NL: PROCEDURE; CALL PR$CHAR( 0DH ); CALL PR$CHAR( 0AH ); END;
PR$SIGNED$NUMBER: PROCEDURE( N ); /* PRINTS A SIGNED NUMBER */
DECLARE N ADDRESS;
DECLARE V ADDRESS, N$STR ( 9 )BYTE, W BYTE;
IF SIGNED$LT( N, 0 ) THEN V = - N; ELSE V = N;
DO W = 0 TO LAST( N$STR ); N$STR( W ) = ' '; END;
W = LAST( N$STR );
N$STR( W ) = '$';
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
IF SIGNED$LT( N, 0 ) THEN N$STR( W := W - 1 ) = '-';
CALL PR$STRING( .N$STR );
END PR$SIGNED$NUMBER;
/* TASK */
DECLARE ( C, N, L, F, Z ) ADDRESS;
DO C = 1 TO 200;
N = C - 100;
L = 0; F = 3; IF SIGNED$LT( N, 0 ) THEN Z = - N; ELSE Z = N;
DO WHILE Z >= 2;
DO WHILE Z MOD 2 = 0; L = L + SIGNED$DIV( N, 2 ); Z = Z / 2; END;
IF F <= Z THEN DO;
DO WHILE Z MOD F = 0; L = L + SIGNED$DIV( N, F ); Z = Z / F; END;
F = F + 2;
END;
END;
CALL PR$SIGNED$NUMBER( L );
IF C MOD 10 = 0 THEN CALL PR$NL;
END;
EOF- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
Phix
with javascript_semantics include mpfr.e procedure D(mpz n) integer s = mpz_cmp_si(n,0) if s<0 then mpz_neg(n,n) end if if mpz_cmp_si(n,2)<0 then mpz_set_si(n,0) else sequence f = mpz_prime_factors(n) integer c = sum(vslice(f,2)), f1 = f[1][1] if c=1 then mpz_set_si(n,1) elsif c=2 then mpz_set_si(n,f1 + iff(length(f)=1?f1:f[2][1])) else assert(mpz_fdiv_q_ui(n,n,f1)=0) mpz d = mpz_init_set(n) D(n) mpz_mul_si(n,n,f1) mpz_add(n,n,d) end if if s<0 then mpz_neg(n,n) end if end if end procedure sequence res = repeat(0,200) mpz n = mpz_init() for i=-99 to 100 do mpz_set_si(n,i) D(n) res[i+100] = mpz_get_str(n) end for printf(1,"%s\n\n",{join_by(res,1,10," ",fmt:="%4s")}) for m=1 to 20 do mpz_ui_pow_ui(n,10,m) D(n) assert(mpz_fdiv_q_ui(n,n,7)=0) printf(1,"D(10^%d)/7 = %s\n",{m,mpz_get_str(n)}) end for
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1)/7 = 1 D(10^2)/7 = 20 D(10^3)/7 = 300 D(10^4)/7 = 4000 D(10^5)/7 = 50000 D(10^6)/7 = 600000 D(10^7)/7 = 7000000 D(10^8)/7 = 80000000 D(10^9)/7 = 900000000 D(10^10)/7 = 10000000000 D(10^11)/7 = 110000000000 D(10^12)/7 = 1200000000000 D(10^13)/7 = 13000000000000 D(10^14)/7 = 140000000000000 D(10^15)/7 = 1500000000000000 D(10^16)/7 = 16000000000000000 D(10^17)/7 = 170000000000000000 D(10^18)/7 = 1800000000000000000 D(10^19)/7 = 19000000000000000000 D(10^20)/7 = 200000000000000000000
Python
from sympy.ntheory import factorint
def D(n):
if n < 0:
return -D(-n)
elif n < 2:
return 0
else:
fdict = factorint(n)
if len(fdict) == 1 and 1 in fdict: # is prime
return 1
return sum([n * e // p for p, e in fdict.items()])
for n in range(-99, 101):
print('{:5}'.format(D(n)), end='\n' if n % 10 == 0 else '')
print()
for m in range(1, 21):
print('(D for 10**{}) divided by 7 is {}'.format(m, D(10 ** m) // 7))
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
(D for 10**1) divided by 7 is 1
(D for 10**2) divided by 7 is 20
(D for 10**3) divided by 7 is 300
(D for 10**4) divided by 7 is 4000
(D for 10**5) divided by 7 is 50000
(D for 10**6) divided by 7 is 600000
(D for 10**7) divided by 7 is 7000000
(D for 10**8) divided by 7 is 80000000
(D for 10**9) divided by 7 is 900000000
(D for 10**10) divided by 7 is 10000000000
(D for 10**11) divided by 7 is 110000000000
(D for 10**12) divided by 7 is 1200000000000
(D for 10**13) divided by 7 is 13000000000000
(D for 10**14) divided by 7 is 140000000000000
(D for 10**15) divided by 7 is 1500000000000000
(D for 10**16) divided by 7 is 16000000000000000
(D for 10**17) divided by 7 is 170000000000000000
(D for 10**18) divided by 7 is 1800000000000000000
(D for 10**19) divided by 7 is 19000000000000000000
(D for 10**20) divided by 7 is 200000000000000000000
Quackery
primefactors is defined at Prime decomposition#Quackery.
[ dup 0 < iff
[ negate
' negate ]
else []
swap 0 over
primefactors
witheach
[ dip over / + ]
nip swap do ] is d ( n --> n )
200 times [ i^ 99 - d echo sp ]
cr cr
20 times [ 10 i^ 1+ ** d 7 / echo cr ]- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 1 20 300 4000 50000 600000 7000000 80000000 900000000 10000000000 110000000000 1200000000000 13000000000000 140000000000000 1500000000000000 16000000000000000 170000000000000000 1800000000000000000 19000000000000000000 200000000000000000000
R
library(gmp) #for big number factorization
arithmetic_derivative<-function(x){
if (x==0|x==1|x==-1){
D=0
}
else{
n=ifelse(x<0,-x,x)
prime_decomposition <-as.numeric(factorize(n))
if (length(prime_decomposition)==1){
D<- 1
}
else{
D<-sum(prime_decomposition[c(1,2)])
if (length(prime_decomposition)>2){
cumulative_prod <-cumprod(prime_decomposition)
for (i in 3:length(prime_decomposition)){
D<- D * prime_decomposition[i] + cumulative_prod[i-1]
}
}
}
}
sign(x)*D
}
print(t(matrix(sapply(-99:100,arithmetic_derivative),nrow=10)))
for (k in 1:20){
x <- 10**k
cat(paste0("D(",x,")/7 = ",arithmetic_derivative(x)/7,"\n"),sep = "")}
Output
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [1,] -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 [2,] -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 [3,] -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 [4,] -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 [5,] -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 [6,] -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 [7,] -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 [8,] -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 [9,] -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 [10,] -6 -12 -1 -5 -1 -4 -1 -1 0 0 [11,] 0 1 1 4 1 5 1 12 6 7 [12,] 1 16 1 9 8 32 1 21 1 24 [13,] 10 13 1 44 10 15 27 32 1 31 [14,] 1 80 14 19 12 60 1 21 16 68 [15,] 1 41 1 48 39 25 1 112 14 45 [16,] 20 56 1 81 16 92 22 31 1 92 [17,] 1 33 51 192 18 61 1 72 26 59 [18,] 1 156 1 39 55 80 18 71 1 176 [19,] 108 43 1 124 22 45 32 140 1 123 [20,] 20 96 34 49 24 272 1 77 75 140 D(10)/7 = 1 D(100)/7 = 20 D(1000)/7 = 300 D(10000)/7 = 4000 D(1e+05)/7 = 50000 D(1e+06)/7 = 6e+05 D(1e+07)/7 = 7e+06 D(1e+08)/7 = 8e+07 D(1e+09)/7 = 9e+08 D(1e+10)/7 = 1e+10 D(1e+11)/7 = 1.1e+11 D(1e+12)/7 = 1.2e+12 D(1e+13)/7 = 1.3e+13 D(1e+14)/7 = 1.4e+14 D(1e+15)/7 = 1.5e+15 D(1e+16)/7 = 1.6e+16 D(1e+17)/7 = 1.7e+17 D(1e+18)/7 = 1.8e+18 D(1e+19)/7 = 1.9e+19 D(1e+20)/7 = 2e+20
Raku
use Prime::Factor;
multi D (0) { 0 }
multi D (1) { 0 }
multi D ($n where &is-prime) { 1 }
multi D ($n where * < 0 ) { -D -$n }
multi D ($n) { sum $n.&prime-factors.Bag.map: { $n × .value / .key } }
put (-99 .. 100).map(&D).batch(10)».fmt("%4d").join: "\n";
put '';
put join "\n", (1..20).map: { sprintf "D(10**%-2d) / 7 == %d", $_, D(10**$_) / 7 }
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10**1 ) / 7 == 1 D(10**2 ) / 7 == 20 D(10**3 ) / 7 == 300 D(10**4 ) / 7 == 4000 D(10**5 ) / 7 == 50000 D(10**6 ) / 7 == 600000 D(10**7 ) / 7 == 7000000 D(10**8 ) / 7 == 80000000 D(10**9 ) / 7 == 900000000 D(10**10) / 7 == 10000000000 D(10**11) / 7 == 110000000000 D(10**12) / 7 == 1200000000000 D(10**13) / 7 == 13000000000000 D(10**14) / 7 == 140000000000000 D(10**15) / 7 == 1500000000000000 D(10**16) / 7 == 16000000000000000 D(10**17) / 7 == 170000000000000000 D(10**18) / 7 == 1800000000000000000 D(10**19) / 7 == 19000000000000000000 D(10**20) / 7 == 200000000000000000000
Refal
$ENTRY Go {
= <Table 10 7 <Each (Lagarias) <Iota ('-'99) 100>>>
};
Lagarias {
'-' e.N = '-' <Lagarias e.N>;
0 = 0;
1 = 0;
e.N, <Fac e.N>: {
e.N = 1;
e.F, <Div (e.N) e.F>: e.R =
<Add <Mul (e.R) <Lagarias e.F>>
<Mul (e.F) <Lagarias e.R>>>;
};
};
Fac {
(e.F) e.N, <Mul (e.N) e.N>: e.N2,
<Compare (e.F) e.N2>: '+' = e.N;
(e.F) e.N, <Mod (e.N) e.F>: 0 = e.F;
(e.F) e.N = <Fac (<Add (1) e.F>) e.N>;
e.N = <Fac (2) e.N>;
};
Table { s.C s.W e.L = <Each (Prout) <Each (Line s.W) <Group s.C e.L>>>; };
Line { s.W e.X = <Join <Each (Fmt s.W) e.X>>; };
Fmt { s.W e.X, <Last s.W <Rep s.W ' '> <Symb e.X>>: (e.Z) e.C = e.C; };
Rep { 0 s.C = ; s.N s.C = s.C <Rep <Sub s.N 1> s.C>; };
Join { = ; (e.X) e.Y = e.X <Join e.Y>; };
Group { s.N = ; s.N e.X, <First s.N e.X>: (e.G) e.R = (e.G) <Group s.N e.R>; };
Each { (e.F) = ; (e.F) (e.X) e.XS = (<Mu e.F e.X>) <Each (e.F) e.XS>; };
Iota { (e.E) e.E = (e.E); (e.S) e.E = (e.S) <Iota (<Add (1) e.S>) e.E>; };- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 -0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
RPL
≪ CASE
DUP 0 < THEN NEG ADERIV NEG END
DUP 2 < THEN DROP 0 END
R→I DUP ISPRIME? THEN DROP 1 END
DUP FACTORS HEAD LASTARG 2 GET DUP2 ^ 4 PICK OVER / 1 RND
→ n p k pk rem
≪ k pk p / * rem * rem ADERIV pk * +
≫
END
≫ 'ADERIV' STO
≪ n ADERIV ≫ 'n' -99 100 1 SEQ ≪ 10 m ^ ADERIV 7 / R→I ≫ 'nm' 1 20 1 SEQ
- Output:
2: {-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140}
1: {1 20 300 4000 50000 600000 7000000 80000000 900000000 10000000000 110000000000 1200000000000 13000000000000 140000000000000 1500000000000000 16000000000000000 170000000000000000 800000000000000000 19000000000000000000 200000000000000000000}
Rust
use prime_factorization::Factorization;
fn d(n: i128) -> i128 {
if n < 0 {
return -(d(-n));
} else if n < 2 {
return 0;
} else {
let fpairs = Factorization::run(n as u128).prime_factor_repr();
if fpairs.len() == 1 && fpairs[0].1 == 1 {
return 1;
}
return fpairs.iter().fold(0_i128, |p, q| p + n * (q.1 as i128) / (q.0 as i128));
}
}
fn main() {
for n in -99..101 {
print!("{:5}{}", d(n), { if n % 10 == 0 { "\n" } else {""} });
}
println!();
for m in 1..21 {
println!("(D for 10 ^ {}) divided by 7 is {}", m, d(10_i128.pow(m)) / 7)
}
}
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
(D for 10 ^ 1) divided by 7 is 1
(D for 10 ^ 2) divided by 7 is 20
(D for 10 ^ 3) divided by 7 is 300
(D for 10 ^ 4) divided by 7 is 4000
(D for 10 ^ 5) divided by 7 is 50000
(D for 10 ^ 6) divided by 7 is 600000
(D for 10 ^ 7) divided by 7 is 7000000
(D for 10 ^ 8) divided by 7 is 80000000
(D for 10 ^ 9) divided by 7 is 900000000
(D for 10 ^ 10) divided by 7 is 10000000000
(D for 10 ^ 11) divided by 7 is 110000000000
(D for 10 ^ 12) divided by 7 is 1200000000000
(D for 10 ^ 13) divided by 7 is 13000000000000
(D for 10 ^ 14) divided by 7 is 140000000000000
(D for 10 ^ 15) divided by 7 is 1500000000000000
(D for 10 ^ 16) divided by 7 is 16000000000000000
(D for 10 ^ 17) divided by 7 is 170000000000000000
(D for 10 ^ 18) divided by 7 is 1800000000000000000
(D for 10 ^ 19) divided by 7 is 19000000000000000000
(D for 10 ^ 20) divided by 7 is 200000000000000000000
Scala
import java.math.BigInteger
object ArithmeticDerivative extends App {
println("Arithmetic derivatives for -99 to 100 inclusive:")
for {
n <- -99 to 100
column = n + 100
} print(f"${derivative(BigInteger.valueOf(n))}%4d${if (column % 10 == 0) "\n" else " "}")
println()
val seven = BigInteger.valueOf(7)
for (power <- 1 to 20) {
println(f"D(10^$power%d) / 7 = ${derivative(BigInteger.TEN.pow(power)).divide(seven)}")
}
def derivative(aNumber: BigInteger): BigInteger = {
if (aNumber.signum == -1) {
return derivative(aNumber.negate()).negate()
}
if (aNumber == BigInteger.ZERO || aNumber == BigInteger.ONE) {
return BigInteger.ZERO
}
var divisor = BigInteger.TWO
while (divisor.multiply(divisor).compareTo(aNumber) <= 0) {
if (aNumber.mod(divisor).signum == 0) {
val quotient = aNumber.divide(divisor)
return quotient.multiply(derivative(divisor)).add(divisor.multiply(derivative(quotient)))
}
divisor = divisor.add(BigInteger.ONE)
}
BigInteger.ONE
}
}
- Output:
Arithmetic derivatives for -99 to 100 inclusive: -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1) / 7 = 1 D(10^2) / 7 = 20 D(10^3) / 7 = 300 D(10^4) / 7 = 4000 D(10^5) / 7 = 50000 D(10^6) / 7 = 600000 D(10^7) / 7 = 7000000 D(10^8) / 7 = 80000000 D(10^9) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
SETL
program arithmetic_derivative;
loop for n in [-99..100] do
nprint(lpad(str lagarias(n), 6));
if (col +:= 1) mod 10 = 0 then
print;
end if;
end loop;
loop for m in [1..20] do
nprint("D(10^" + lpad(str m, 2) + ") / 7 = ");
print(lagarias(10**m) div 7);
end loop;
proc lagarias(n);
return if n<0 then
-lagarias(-n)
elseif n in {0,1} then
0
elseif forall d in {2..floor sqrt n} | n mod d /= 0 then
1
else
(n div d)*lagarias(d) + d*lagarias(n div d)
end;
end proc;
end program;- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
D(10^ 1) / 7 = 1
D(10^ 2) / 7 = 20
D(10^ 3) / 7 = 300
D(10^ 4) / 7 = 4000
D(10^ 5) / 7 = 50000
D(10^ 6) / 7 = 600000
D(10^ 7) / 7 = 7000000
D(10^ 8) / 7 = 80000000
D(10^ 9) / 7 = 900000000
D(10^10) / 7 = 10000000000
D(10^11) / 7 = 110000000000
D(10^12) / 7 = 1200000000000
D(10^13) / 7 = 13000000000000
D(10^14) / 7 = 140000000000000
D(10^15) / 7 = 1500000000000000
D(10^16) / 7 = 16000000000000000
D(10^17) / 7 = 170000000000000000
D(10^18) / 7 = 1800000000000000000
D(10^19) / 7 = 19000000000000000000
D(10^20) / 7 = 200000000000000000000
Sidef
Built-in as Number#arithmetic_derivative:
say "Arithmetic derivative for n in range [-99, 100]:"
-99 .. 100 -> map { .arithmetic_derivative }.each_slice(10, {|*a|
a.map { '%4s' % _ }.join(' ').say
})
say "\nArithmetic derivative D(10^n)/7 for n in range [1, 20]:"
for n in (1..20) {
say "D(10^#{n})/7 = #{arithmetic_derivative(10**n) / 7}"
}
- Output:
Arithmetic derivative for n in range [-99, 100]: -75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 Arithmetic derivative D(10^n)/7 for n in range [1, 20]: D(10^1)/7 = 1 D(10^2)/7 = 20 D(10^3)/7 = 300 D(10^4)/7 = 4000 D(10^5)/7 = 50000 D(10^6)/7 = 600000 D(10^7)/7 = 7000000 D(10^8)/7 = 80000000 D(10^9)/7 = 900000000 D(10^10)/7 = 10000000000 D(10^11)/7 = 110000000000 D(10^12)/7 = 1200000000000 D(10^13)/7 = 13000000000000 D(10^14)/7 = 140000000000000 D(10^15)/7 = 1500000000000000 D(10^16)/7 = 16000000000000000 D(10^17)/7 = 170000000000000000 D(10^18)/7 = 1800000000000000000 D(10^19)/7 = 19000000000000000000 D(10^20)/7 = 200000000000000000000
Explicit implementation:
subset Integer < Number { .is_int }
subset Positive < Integer { .is_pos }
subset Negative < Integer { .is_neg }
subset Prime < Positive { .is_prime }
func arithmetic_derivative((0)) { 0 }
func arithmetic_derivative((1)) { 0 }
func arithmetic_derivative(Prime _) { 1 }
func arithmetic_derivative(Negative n) {
-arithmetic_derivative(-n)
}
func arithmetic_derivative(Positive n) is cached {
var a = n.factor.rand
var b = n/a
arithmetic_derivative(a)*b + a*arithmetic_derivative(b)
}
func arithmetic_derivative(Number n) {
var (a, b) = n.nude
(arithmetic_derivative(a)*b - arithmetic_derivative(b)*a) / b**2
}
Wren
As integer arithmetic in Wren is inaccurate above 2^53 we need to use BigInt here.
import "./big" for BigInt
import "./fmt" for Fmt
var D = Fn.new { |n|
if (n < 0) return -D.call(-n)
if (n < 2) return BigInt.zero
var f = BigInt.primeFactors(n)
var c = f.count
if (c == 1) return BigInt.one
if (c == 2) return f[0] + f[1]
var d = n / f[0]
return D.call(d) * f[0] + d
}
var ad = List.filled(200, 0)
for (n in -99..100) ad[n+99] = D.call(BigInt.new(n))
Fmt.tprint("$4i", ad, 10)
System.print()
for (m in 1..20) {
Fmt.print("D(10^$-2d) / 7 = $i", m, D.call(BigInt.ten.pow(m))/7)
}
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10^1 ) / 7 = 1 D(10^2 ) / 7 = 20 D(10^3 ) / 7 = 300 D(10^4 ) / 7 = 4000 D(10^5 ) / 7 = 50000 D(10^6 ) / 7 = 600000 D(10^7 ) / 7 = 7000000 D(10^8 ) / 7 = 80000000 D(10^9 ) / 7 = 900000000 D(10^10) / 7 = 10000000000 D(10^11) / 7 = 110000000000 D(10^12) / 7 = 1200000000000 D(10^13) / 7 = 13000000000000 D(10^14) / 7 = 140000000000000 D(10^15) / 7 = 1500000000000000 D(10^16) / 7 = 16000000000000000 D(10^17) / 7 = 170000000000000000 D(10^18) / 7 = 1800000000000000000 D(10^19) / 7 = 19000000000000000000 D(10^20) / 7 = 200000000000000000000
XPL0
function integer Lagarias (N); \Lagarias arithmetic derivative
integer N;
integer F, Q;
function integer SmallPF (J, K); \Smallest prime factor
integer J, K;
return if rem(J/K) = 0 then K else SmallPF(J, K+1);
begin
if N < 0
then return -Lagarias (-N)
else if N = 0 or N = 1
then return 0
else begin
F := SmallPF (N, 2); Q := N / F;
return if Q = 1
then 1
else Q * Lagarias (F) + F * Lagarias (Q)
end;
end \Lagarias\ ;
integer N;
begin
for N:= -99 to 100 do begin
IntOut(0, Lagarias(N) );
if rem(N/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
end;
end- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140
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