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.

0, 7, 8, 12, 42

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..5.

Onno M. Cain, Bioperational Multisets in Various Semi-rings, arXiv:1908.03235 [math.RA], 2019.

Michael W. Ecker, When Does a Sum of Positive Integers Equal Their Product? Mathematics Magazine 75(1), 2002, pp. 41-47.

Michael A. Nyblom, Sophie Germain Primes and the Exceptional Values of the Equal-Sum-And-Product Problem, Fib. Q. 50(1), 2012, 58-61.

Example

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.

Crossrefs

Cf. A343297, A033178, A033179.

Sequence in context: A059177 A065729 A156605 * A061905 A308887 A308833

Adjacent sequences: A343295 A343296 A343297 * A343299 A343300 A343301

Keyword

nonn,hard,more

Author

Nathaniel Gregg, Apr 11 2021

Status

approved