A341866 The cardinality of the smallest (nontrivial, except for prime n) multiset of positive integers whose product and sum equal n.

1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 1, 6, 1, 7, 9, 8, 1, 9, 1, 10, 13, 11, 1, 12, 17, 13, 17, 14, 1, 15, 1, 16, 21, 17, 25, 18, 1, 19, 25, 20, 1, 21, 1, 22, 29, 23, 1, 24, 37, 25, 33, 26, 1, 27, 41, 28, 37, 29, 1, 30, 1, 31, 41, 32

Offset

1,4

Comments

The smallest set is obtained by taking the largest such multiset (A341865(n)) and replacing the largest proper subset that is also a product-sum multiset with its product. A singleton would always be the smallest product-sum multiset, so those are excluded except for prime n where no nontrivial multisets exist.

Links

Table of n, a(n) for n=1..64.

Formula

a(n) = (n/p - 1)*(p-1) + 1, where p is the smallest factor of n.

a(n) = A341865(n) - A341865(n/p) + 1, where p is the smallest prime factor of n.

Example

For n = 12, the set of size a(n) = 6 is {1,1,1,1,2,6}.

Prog

(PARI) a(n) = if (n==1, 1, my(p=vecmin(factor(n)[, 1])); (n/p-1)*(p-1) + 1); \\ Michel Marcus, Feb 26 2021

Crossrefs

Cf. A104173, A033178, A033179, A341865.

Equals A330492 + 1. - Hugo Pfoertner, Feb 23 2021

Sequence in context: A142878 A299774 A362254 * A300242 A293892 A295885

Adjacent sequences: A341863 A341864 A341865 * A341867 A341868 A341869

Keyword

nonn

Author

Nathaniel Gregg, Feb 22 2021

Status

approved