A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.

1, 1, 2, 2, 5, 4, 11, 10, 21, 22, 51, 40, 106, 103, 200, 205, 465, 405, 961, 900, 1828, 1913, 4096, 3578, 8273, 8175, 16132, 16267, 34903, 31822, 70854, 68681, 137391, 140661, 292081, 270258, 591443, 582453, 1156012

Offset: 1

Author

P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006

Keywords

• nonn
• more

Comments

From Gus Wiseman, Aug 28 2023: (Start)

Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:

{1} {2} {3} {4} {5} {6} {7} {8}

{2,3} {3,4} {2,5} {4,6} {2,7} {3,8}

{3,5} {5,6} {3,7} {5,8}

{4,5} {4,5,6} {4,7} {6,8}

{3,4,5} {5,7} {7,8}

{6,7} {3,7,8}

{3,5,7} {5,6,8}

{4,5,7} {5,7,8}

{4,6,7} {6,7,8}

{5,6,7} {5,6,7,8}

{4,5,6,7}

Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.

(End)

Examples

			a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.
a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.
a(3) = 2 via <2,5> = {0,2,4,5,...}; and <4,5,6,7> = {0,4,5,6,7,...} where in both the largest missing number is 3.
a(4) = 2 via <3,5,7> = {0,3,5,6,7,...} and <5,6,7,8,9> = {5,6,7,8,9,...} where in both the largest missing number is 4.
		

Links

• S. R. Finch, Monoids of natural numbers
• Manuel Delgado, Neeraj Kumar, and Claude Marion, On counting numerical semigroups by maximum primitive and Wilf's conjecture, arXiv:2501.04417 [math.CO], 2025. See p. 22.
• S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
• J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia, and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
• Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).

Crossrefs

Cf. A158206. [From Steven Finch, Mar 13 2009]

A288728 counts sum-free sets, first differences of A007865.

A364350 counts combination-free partitions, complement A364839.

Cf. A085489, A088809, A093971, A103580, A116861, A151897, A237668, A308546, A326020, A326083, A364349, A365069.

Programs

GAP

The sequence was originally generated by a C program and a Haskell script. The sequence can be obtained by using the function NumericalSemigroupsWithFrobeniusNumber included in the numericalsgps GAP package.