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

1,3

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)

Links

Table of n, a(n) for n=1..39.

S. R. Finch, Monoids of natural numbers

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).

Example

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.

Prog

(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.

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.

Sequence in context: A292263 A338528 A238624 * A264687 A112471 A144366

Adjacent sequences: A124503 A124504 A124505 * A124507 A124508 A124509

Keyword

nonn,more

Author

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

Status

approved