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A055977 Numbers k such that Product_{q|k} p(q) divides p(k), where p(k) is number of unrestricted partitions of k and the product is over all distinct primes q that divide k.

%I A055977 #18 Jul 25 2024 03:44:24
%S A055977 1,2,3,5,7,8,9,10,11,13,17,19,23,29,31,37,40,41,43,47,53,59,61,64,67,
%T A055977 71,73,75,79,83,89,97,101,103,107,109,113,125,127,128,131,137,139,149,
%U A055977 151,157,163,167,173,179,180,181,189,191,193,197,199,211,223,225,227
%N A055977 Numbers k such that Product_{q|k} p(q) divides p(k), where p(k) is number of unrestricted partitions of k and the product is over all distinct primes q that divide k.
%H A055977 John Tyler Rascoe, <a href="/A055977/b055977.txt">Table of n, a(n) for n = 1..2000</a>
%e A055977 10 is included because p(10) = 42 is divisible by p(2)*p(5) = 2*7 and 2 and 5 are the distinct prime divisors of 10.
%o A055977 (Python)
%o A055977 from itertools import count, islice
%o A055977 from math import prod
%o A055977 from sympy.ntheory import npartitions, factorint
%o A055977 def a_gen():
%o A055977     for n in count(1):
%o A055977         if npartitions(n)%prod([npartitions(i) for i in factorint(n)]) < 1:
%o A055977             yield n
%o A055977 A055977_list = list(islice(a_gen(), 61)) # _John Tyler Rascoe_, Jul 24 2024
%o A055977 (PARI) isok(k) = my(f=factor(k)); numbpart(k) % prod(i=1, #f~, numbpart(f[i,1])) == 0; \\ _Michel Marcus_, Jul 25 2024
%Y A055977 Cf. A000041, A054411.
%K A055977 easy,nonn
%O A055977 1,2
%A A055977 _Leroy Quet_, Jul 20 2000
%E A055977 Name and offset edited by _John Tyler Rascoe_, Jul 24 2024