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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.
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%I #18 Jul 25 2024 03:44:24

%S 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 71,73,75,79,83,89,97,101,103,107,109,113,125,127,128,131,137,139,149,

%U 151,157,163,167,173,179,180,181,189,191,193,197,199,211,223,225,227

%N 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 John Tyler Rascoe, <a href="/A055977/b055977.txt">Table of n, a(n) for n = 1..2000</a>

%e 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 (Python)

%o from itertools import count, islice

%o from math import prod

%o from sympy.ntheory import npartitions, factorint

%o def a_gen():

%o for n in count(1):

%o if npartitions(n)%prod([npartitions(i) for i in factorint(n)]) < 1:

%o yield n

%o A055977_list = list(islice(a_gen(), 61)) # _John Tyler Rascoe_, Jul 24 2024

%o (PARI) isok(k) = my(f=factor(k)); numbpart(k) % prod(i=1, #f~, numbpart(f[i,1])) == 0; \\ _Michel Marcus_, Jul 25 2024

%Y Cf. A000041, A054411.

%K easy,nonn

%O 1,2

%A _Leroy Quet_, Jul 20 2000

%E Name and offset edited by _John Tyler Rascoe_, Jul 24 2024