Calmo numbers
Let n be a natural number having k divisors (other than 1 and n itself) where k is exactly divisible by 3.
Calmo numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
- Definition
Add the first three eligible divisors, then the next three, and so on until the eligible divisors are exhausted. If the resulting partial sums are prime numbers, then n is called a Calmo number.
- Example
Consider n = 165.
It has 6 eligible divisors, namely [3 5 11 15 33 55].
The sum of the first three is: 3 + 5 + 11 = 19 which is a prime number.
The sum of the next three is: 15 + 33 + 55 = 103 which is also a prime number.
Hence n is a Calmo number.
- Task
Find and show here all Calmo numbers under 1000.
ALGOL 68
Note, the source of primes.incl.a68 is on Rosetta Code (see above link).
BEGIN # find some "Calmo" numbers: numbers n such that they have 3k divisors #
# (other than 1 and n) for some k > 0 and the sum of their divisors #
# taken three at a time is a prime #
PR read "primes.incl.a68" PR # include prime utilities #
INT max number = 1 000; # largest number we will consider #
# construct a sieve of (hopefully) enough primes - as we are going to sum #
# the divisors in groups of three, it should be (more than) large enough #
[]BOOL prime = PRIMESIEVE ( max number * 3 );
# construct tables of the divisor counts and divisor sums and check for #
# the numbers as we do it #
# as we are ignoring 1 and n, the initial counts and sums will be 0 #
# but we should ignore primes #
[ 1 : max number ]INT dsum;
[ 1 : max number ]INT dcount;
FOR i TO UPB dcount DO
dsum[ i ] := dcount[ i ] := IF prime[ i ] THEN -1 ELSE 0 FI
OD;
FOR i FROM 2 TO UPB dsum
DO FOR j FROM i + i BY i TO UPB dsum DO
# have another proper divisor #
IF dsum[ j ] >= 0 THEN
# this number is still a candidate #
dsum[ j ] +:= i;
dcount[ j ] +:= 1;
IF dcount[ j ] = 3 THEN
# the divisor count is currently 3 #
dsum[ j ] := dcount[ j ] :=
IF NOT prime[ dsum[ j ] ] THEN
# divisor sum isn't prime, ignore it in future #
-1
ELSE
# divisor count is prime, reset the sum and count #
0
FI
FI
FI
OD
OD;
# show the numbers #
FOR i FROM 2 TO UPB dcount DO
IF dcount[ i ] = 0 THEN
# have a number #
print( ( " ", whole( i, 0 ) ) )
FI
OD;
print( ( newline ) )
END- Output:
165 273 385 399 561 595 665 715 957
Raku
use Prime::Factor;
use List::Divvy;
my $upto = 1e3;
my @found = (2..Inf).hyper.grep({
(so my @d = .&proper-divisors(:s)) &&
(@d.elems %% 3) &&
(all @d.batch(3)».sum».is-prime)
}).&upto($upto);
put "{+@found} found before $upto using sums of proper-divisors:\n" ~
@found.batch(10)».fmt("%4d").join: "\n";
@found = (2..Inf).hyper.grep({
(so my @d = .&proper-divisors(:s).&after: 1) &&
(@d.elems %% 3) &&
(all @d.batch(3)».sum».is-prime)
}).&upto($upto);
put "\n{+@found} found before $upto using sums of some bizarre\nbespoke definition for divisors:\n" ~
@found.batch(10)».fmt("%4d").join: "\n";
- Output:
85 found before 1000 using sums of proper-divisors: 8 21 27 35 39 55 57 65 77 85 111 115 125 129 155 161 185 187 201 203 205 209 221 235 237 265 291 299 305 309 319 323 327 335 341 365 371 377 381 391 413 415 437 451 485 489 493 497 505 515 517 535 579 611 623 649 655 667 669 671 687 689 697 707 731 737 755 767 779 781 785 831 835 851 865 893 899 901 917 921 939 955 965 979 989 9 found before 1000 using sums of some bizarre bespoke definition for divisors: 165 273 385 399 561 595 665 715 957
Ring
see "works..." + nl
numCalmo = 0
limit = 1000
for n = 1 to limit
Calmo = []
for m = 2 to n/2
if n % m = 0
add(Calmo,m)
ok
next
flag = 1
lenCalmo = len(Calmo)
if (lenCalmo > 5) and (lenCalmo % 3 = 0)
for p = 1 to lenCalmo - 2 step 3
sum = Calmo[p] + Calmo[p+1] + Calmo[p+2]
if not isPrime(sum)
flag = 0
exit
ok
next
if flag = 1
numCalmo++
see "n(" + numCalmo + ") = " + n + nl
see "divisors = ["
for p = 1 to lenCalmo - 2 step 3
sumCalmo = Calmo[p] + Calmo[p+1] + Calmo[p+2]
if not isPrime(sumCalmo)
exit
else
if p = 1
see "" + Calmo[p] + " " + Calmo[p+1] + " " + Calmo[p+2]
else
see " " + Calmo[p] + " " + Calmo[p+1] + " " + Calmo[p+2]
ok
ok
next
see "]" + nl
for p = 1 to lenCalmo - 2 step 3
sumCalmo = Calmo[p] + Calmo[p+1] + Calmo[p+2]
if isPrime(sumCalmo)
see "" + Calmo[p] + " + " + Calmo[p+1] + " + " + Calmo[p+2] + " = " + sumCalmo + " is prime" + nl
ok
next
see nl
ok
ok
next
see "Found " + numCalmo + " Calmo numbers" + nl
see "done..." + nl
func isPrime num
if (num <= 1) return 0 ok
if (num % 2 = 0 and num != 2) return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1- Output:
works... n(1) = 165 divisors = [3 5 11 15 33 55] 3 + 5 + 11 = 19 is prime 15 + 33 + 55 = 103 is prime n(2) = 273 divisors = [3 7 13 21 39 91] 3 + 7 + 13 = 23 is prime 21 + 39 + 91 = 151 is prime n(3) = 385 divisors = [5 7 11 35 55 77] 5 + 7 + 11 = 23 is prime 35 + 55 + 77 = 167 is prime n(4) = 399 divisors = [3 7 19 21 57 133] 3 + 7 + 19 = 29 is prime 21 + 57 + 133 = 211 is prime n(5) = 561 divisors = [3 11 17 33 51 187] 3 + 11 + 17 = 31 is prime 33 + 51 + 187 = 271 is prime n(6) = 595 divisors = [5 7 17 35 85 119] 5 + 7 + 17 = 29 is prime 35 + 85 + 119 = 239 is prime n(7) = 665 divisors = [5 7 19 35 95 133] 5 + 7 + 19 = 31 is prime 35 + 95 + 133 = 263 is prime n(8) = 715 divisors = [5 11 13 55 65 143] 5 + 11 + 13 = 29 is prime 55 + 65 + 143 = 263 is prime n(9) = 957 divisors = [3 11 29 33 87 319] 3 + 11 + 29 = 43 is prime 33 + 87 + 319 = 439 is prime Found 9 Calmo numbers done...
Wren
import "./math" for Int, Nums
import "./seq" for Lst
import "./fmt" for Fmt
var limit = 1000
var calmo = []
for (i in 2...limit) {
var ed = Int.properDivisors(i)
ed.removeAt(0)
if (ed.count == 0 || ed.count % 3 != 0) continue
var isCalmo = true
var ps = []
for (chunk in Lst.chunks(ed, 3)) {
var sum = Nums.sum(chunk)
if (!Int.isPrime(sum)) {
isCalmo = false
break
}
ps.add(sum)
}
if (isCalmo) calmo.add([i, ed, ps])
}
System.print("Calmo numbers under 1,000:\n")
System.print("Number Eligible divisors Partial sums")
System.print("----------------------------------------------")
for (e in calmo) {
Fmt.print("$3d $-24n $n", e[0], e[1], e[2])
}- Output:
Calmo numbers under 1,000: Number Eligible divisors Partial sums ---------------------------------------------- 165 [3, 5, 11, 15, 33, 55] [19, 103] 273 [3, 7, 13, 21, 39, 91] [23, 151] 385 [5, 7, 11, 35, 55, 77] [23, 167] 399 [3, 7, 19, 21, 57, 133] [29, 211] 561 [3, 11, 17, 33, 51, 187] [31, 271] 595 [5, 7, 17, 35, 85, 119] [29, 239] 665 [5, 7, 19, 35, 95, 133] [31, 263] 715 [5, 11, 13, 55, 65, 143] [29, 263] 957 [3, 11, 29, 33, 87, 319] [43, 439]