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A343298 a(n) is the smallest m such that the only non-basic multiset of positive integers of cardinality m where the sum equals the product has n nonunit elements, or zero if no such m exists.

%I A343298 #18 Jan 05 2025 19:51:42
%S A343298 0,7,8,12,42
%N A343298 a(n) is the smallest m such that the only non-basic multiset of positive integers of cardinality m where the sum equals the product has n nonunit elements, or zero if no such m exists.
%C A343298 A multiset where the sum equals the product is called a bioperational multiset. A bioperational multiset is called basic if it is of the form {2,n,1,...,1}, because these exist of size n for all n > 1. Nonzero entries are a subset of A343297. a(6) > 10^4 or zero.
%H A343298 Onno M. Cain, <a href="https://arxiv.org/abs/1908.03235">Bioperational Multisets in Various Semi-rings</a>, arXiv:1908.03235 [math.RA], 2019.
%H A343298 Michael W. Ecker, <a href="http://www.jstor.org/stable/3219187">When Does a Sum of Positive Integers Equal Their Product?</a> Mathematics Magazine 75(1), 2002, pp. 41-47.
%H A343298 Michael A. Nyblom, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/50-1/Nyblom.pdf">Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem</a>, Fib. Q. 50(1), 2012, 58-61.
%e A343298 a(5) = 42 because {2,2,2,2,3; 37} and {42,2; 40} are the only bioperational multisets of size 42, where the number after the semicolon is the number of repeated 1's.
%Y A343298 Cf. A343297, A033178, A033179.
%K A343298 nonn,hard,more
%O A343298 1,2
%A A343298 _Nathaniel Gregg_, Apr 11 2021