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

1,2

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).

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to partitions

Example

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.

Prog

(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

Crossrefs

Cf, A000005, A000041, A000203, A018818.

Sequence in context: A286376 A100346 A359004 * A308692 A319352 A300834

Adjacent sequences: A306384 A306385 A306386 * A306388 A306389 A306390

Keyword

nonn

Author

Marius A. Burtea, Feb 26 2019

Extensions

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

Status

approved