A341865 The cardinality of the largest multiset of positive integers whose product and sum equals n.

1, 1, 1, 2, 1, 3, 1, 5, 5, 5, 1, 8, 1, 7, 9, 12, 1, 13, 1, 14, 13, 11, 1, 19, 17, 13, 21, 20, 1, 23, 1, 27, 21, 17, 25, 30, 1, 19, 25, 33, 1, 33, 1, 32, 37, 23, 1, 42, 37, 41, 33, 38, 1, 47, 41, 47, 37, 29, 1, 52, 1, 31, 53, 58

Offset: 1

Author

Nathaniel Gregg, Feb 22 2021

Keywords

• nonn

Comments

The largest multisets are given by the prime factorization of n and 1s added until the sum equals the product.

Examples

			For n = 12, the set of size a(12) = 8 is {1,1,1,1,1,2,2,3}.
		

Crossrefs

Cf. A059975, A104173, A033178, A033179, A341866.

Programs

PARI

a(n) = my(f=factor(n)); n - sum(k=1, #f~, f[k,2]*(f[k,1]-1)); \\ Michel Marcus, Feb 26 2021

Formula

a(n) = n - Sum_(d_i*(p_i-1)), where n = Product_(p_i^d_i).

a(n) = n - A059975(n). - Joerg Arndt, Feb 22 2021