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The cardinality of the largest multiset of positive integers whose product and sum equals n.
2

%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