Calmo numbers
Definition Let n be a natural number. Let the real divisors of n be: d(1),d(2),d(3),...,d(k), where k is divisible by 3. Add the first three divisors, then the next three, and so on. If the partial sums are prime numbers, then n is called a Calmo number. Example n = 165 divisors = [3 5 11 15 33 55] 3 + 5 + 11 = 19 is prime number 15 + 33 + 55 = 103 is prime number Task let's find Calmo numbers under 1000.
ALGOL 68
Note, the source of primes.incl.a68 is on Rosetta Code (see above link).
Assuming the task author intended that 1 and n should be excluded when considering the divisors of n,
the output confirms the task author's original sample which they deleted for some reason.
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