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

1,2

Links

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

Example

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.

Prog

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

Crossrefs

Cf. A000041, A054411.

Sequence in context: A047490 A357065 A039072 * A180221 A111292 A047373

Adjacent sequences: A055974 A055975 A055976 * A055978 A055979 A055980

Keyword

easy,nonn

Author

Leroy Quet, Jul 20 2000

Extensions

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

Status

approved