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

1,4

Comments

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

Links

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

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

Example

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

Prog

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

Crossrefs

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

Sequence in context: A065548 A152063 A336617 * A347982 A321738 A022458

Adjacent sequences: A341862 A341863 A341864 * A341866 A341867 A341868

Keyword

nonn

Author

Nathaniel Gregg, Feb 22 2021

Status

approved