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.

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, 71, 73, 75, 79, 83, 89, 97, 101, 103, 107, 109, 113, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223, 225, 227

Offset: 1

Author

Leroy Quet, Jul 20 2000

Keywords

• easy
• nonn

Examples

			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.
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000041, A054411.

Programs

PARI

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

Python

from itertools import count, islice
from math import prod
from sympy.ntheory import npartitions, factorint
def a_gen():
    for n in count(1):
        if npartitions(n)%prod([npartitions(i) for i in factorint(n)]) < 1:
            yield n
A055977_list = list(islice(a_gen(), 61)) # John Tyler Rascoe, Jul 24 2024

Extensions

Name and offset edited by John Tyler Rascoe, Jul 24 2024