A306387 Number of partitions of sigma(n) into divisors of n, where sigma = A000203.

1, 2, 2, 6, 2, 27, 2, 26, 7, 31, 2, 574, 2, 38, 33, 166, 2, 879, 2, 924, 39, 52, 2, 23732, 9, 59, 47, 1403, 2, 34256, 2, 1626, 55, 73, 47, 230819, 2, 80, 61, 50888, 2, 65638, 2, 2709, 1734, 94, 2, 2117920, 11, 3038, 77, 3536, 2, 113448, 65, 97298, 83, 115, 2, 92531888, 2, 122, 2601, 25510, 73, 180350

Offset: 1

Author

Marius A. Burtea, Feb 26 2019

Keywords

• nonn

Comments

The equality sigma(n) = Sum{d|n} d defines unique partition of sigma(n) into distinct divisors of n. This sequence gives the number of partitions of sigma(n) into not necessarily distinct divisors of n.

For prime number p, sigma(p) = p+1 and there are only two partitions: p and 1+1+1+...+1 (p summands).

Examples

			For n = 4, sigma(4) = 7, divisors(4) = {1,2,4} and 7 = 4+2+1 = 4+1+1+1 = 2+2+2+1 = 2+2+1+1+1 = 2+1+1+1+1+1 = 1+1+1+1+1+1+1.
For n = 9, sigma(9) = 13, divisors(9) = {1,3,9} and 13 = 9+3+1 = 9+1+1+1+1 = 3+3+3+3+1 = 3+3+3+1+1+1+1 = 3+3+1+1+1+1+1+1+1 = 3+1+1+1+1+1+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1+1+1.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences related to partitions

Crossrefs

Cf, A000005, A000041, A000203, A018818.

Programs

Magma

v:=[1..47];
for u in v do
u, #RestrictedPartitions(SumOfDivisors(u),{d:d in Divisors(u)});
end for;

Magma

a:= func< n | #RestrictedPartitions(SumOfDivisors(n),{d:d in Divisors(n)}) >; [ a(n) : n in [1..47] ];

PARI

numbpartUsing(n, v, mx=#v)=if(n<1, return(n==0)); sum(i=1, mx, numbpartUsing(n-v[i], v, i)) \\ inefficient;
a(n) = numbpartUsing(sigma(n), divisors(n)); \\ after A018818; Michel Marcus, Feb 27 2019

PARI

A306387(n) = { my(p=1, s=sigma(n)); fordiv(n, d, p /= (1 - 'x^d)); polcoeff(Ser(p,'x,1+s), s); }; \\ Antti Karttunen, Jan 22 2025

Extensions

Term a(60) corrected from 19613170 to 92531888 by Antti Karttunen, Jan 22 2025