Nathaniel Gregg - cp's OEIS frontend

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

Nathaniel Gregg, Apr 11 2021

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.

			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.

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.

Cf. A343297, A033178, A033179.

A343297 Numbers k such that there are exactly two multisets of cardinality k where the sum equals the product (A033178(k)=2).

Original entry on oeis.org

7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 22, 30, 34, 36, 42, 44, 48, 54, 60, 66, 80, 84, 90, 112, 126, 142, 192, 210, 234, 252, 258, 330, 350, 354, 440, 594, 654, 714, 720, 780, 966, 1102, 2400, 2820, 4350, 4354, 5274, 6174, 6324

Offset: 1

Nathaniel Gregg, Apr 11 2021

nonn

At most one of a(n) - 1 and 2*a(n)-1 are composite. More precisely, a(n) are those positive integers such that exactly one of product(s)*(a(n)+sum(s)-k-2)+1 can be factored as (product(s)*p-1)*(product(s)*q-1), where s varies over all multisets of k positive integers and 1 < p <= q < a(n). The first statement is given by considering s = {} and s = {2}. a(50) is greater than 10^4.

			a(5) = 12 because {2,2,2,2,1,1,1,1,1,1,1,1} and {12,2,1,1,1,1,1,1,1,1,1,1} are the only multisets of size 12 where the sum equals the product.

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

Cf. A033178, A033179, A343298.

A341866 The cardinality of the smallest (nontrivial, except for prime n) multiset of positive integers whose product and sum equal n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 5, 5, 1, 6, 1, 7, 9, 8, 1, 9, 1, 10, 13, 11, 1, 12, 17, 13, 17, 14, 1, 15, 1, 16, 21, 17, 25, 18, 1, 19, 25, 20, 1, 21, 1, 22, 29, 23, 1, 24, 37, 25, 33, 26, 1, 27, 41, 28, 37, 29, 1, 30, 1, 31, 41, 32

Offset: 1

Nathaniel Gregg, Feb 22 2021

nonn

The smallest set is obtained by taking the largest such multiset (A341865(n)) and replacing the largest proper subset that is also a product-sum multiset with its product. A singleton would always be the smallest product-sum multiset, so those are excluded except for prime n where no nontrivial multisets exist.

			For n = 12, the set of size a(n) = 6 is {1,1,1,1,2,6}.

Cf. A104173, A033178, A033179, A341865.

Equals A330492 + 1. - Hugo Pfoertner, Feb 23 2021

a(n) = if (n==1, 1, my(p=vecmin(factor(n)[,1])); (n/p-1)*(p-1) + 1); \\ Michel Marcus, Feb 26 2021

a(n) = (n/p - 1)*(p-1) + 1, where p is the smallest factor of n.

a(n) = A341865(n) - A341865(n/p) + 1, where p is the smallest prime factor of n.

A341865 The cardinality of the largest multiset of positive integers whose product and sum equals n.

Original entry on oeis.org

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

Nathaniel Gregg, Feb 22 2021

nonn

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

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

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

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

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

cp's OEIS Frontend

User: Nathaniel Gregg

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A343297 Numbers k such that there are exactly two multisets of cardinality k where the sum equals the product (A033178(k)=2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A341866 The cardinality of the smallest (nontrivial, except for prime n) multiset of positive integers whose product and sum equal n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A341865 The cardinality of the largest multiset of positive integers whose product and sum equals n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula