Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #16 Mar 23 2021 04:53:14
%S 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,
%T 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,
%U 41,47,37,29,1,52,1,31,53,58
%N The cardinality of the largest multiset of positive integers whose product and sum equals n.
%C The largest multisets are given by the prime factorization of n and 1s added until the sum equals the product.
%F a(n) = n - Sum_(d_i*(p_i-1)), where n = Product_(p_i^d_i).
%F a(n) = n - A059975(n). - _Joerg Arndt_, Feb 22 2021
%e For n = 12, the set of size a(12) = 8 is {1,1,1,1,1,2,2,3}.
%o (PARI) a(n) = my(f=factor(n)); n - sum(k=1, #f~, f[k,2]*(f[k,1]-1)); \\ _Michel Marcus_, Feb 26 2021
%Y Cf. A059975, A104173, A033178, A033179, A341866.
%K nonn
%O 1,4
%A _Nathaniel Gregg_, Feb 22 2021