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

%I A124506 #33 Jan 12 2025 10:37:10
%S A124506 1,1,2,2,5,4,11,10,21,22,51,40,106,103,200,205,465,405,961,900,1828,
%T A124506 1913,4096,3578,8273,8175,16132,16267,34903,31822,70854,68681,137391,
%U A124506 140661,292081,270258,591443,582453,1156012
%N A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.
%C A124506 From _Gus Wiseman_, Aug 28 2023: (Start)
%C A124506 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:
%C A124506   {1}  {2}  {3}    {4}    {5}      {6}      {7}        {8}
%C A124506             {2,3}  {3,4}  {2,5}    {4,6}    {2,7}      {3,8}
%C A124506                           {3,5}    {5,6}    {3,7}      {5,8}
%C A124506                           {4,5}    {4,5,6}  {4,7}      {6,8}
%C A124506                           {3,4,5}           {5,7}      {7,8}
%C A124506                                             {6,7}      {3,7,8}
%C A124506                                             {3,5,7}    {5,6,8}
%C A124506                                             {4,5,7}    {5,7,8}
%C A124506                                             {4,6,7}    {6,7,8}
%C A124506                                             {5,6,7}    {5,6,7,8}
%C A124506                                             {4,5,6,7}
%C A124506 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.
%C A124506 (End)
%H A124506 S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a>
%H A124506 Manuel Delgado, Neeraj Kumar, and Claude Marion, <a href="https://arxiv.org/abs/2501.04417">On counting numerical semigroups by maximum primitive and Wilf's conjecture</a>, arXiv:2501.04417 [math.CO], 2025. See p. 22.
%H A124506 S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009. [Cached copy, with permission of the author]
%H A124506 J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia, and J. A. Jimenez-Madrid, <a href="https://doi.org/10.1016/j.jpaa.2003.10.024">Fundamental gaps in numerical semigroups</a>, Journal of Pure and Applied Algebra 189 (2004) 301-313.
%H A124506 Clayton Cristiano Silva, <a href="https://web.archive.org/web/20221006031931/http://www.ime.unicamp.br/~ftorres/ENSINO/MONOGRAFIAS/Clayton.pdf">Irreducible Numerical Semigroups</a>, University of Campinas, São Paulo, Brazil (2019).
%e A124506 a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.
%e A124506 a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.
%e A124506 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.
%e A124506 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.
%o A124506 (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.
%Y A124506 Cf. A158206. [From _Steven Finch_, Mar 13 2009]
%Y A124506 A288728 counts sum-free sets, first differences of A007865.
%Y A124506 A364350 counts combination-free partitions, complement A364839.
%Y A124506 Cf. A085489, A088809, A093971, A103580, A116861, A151897, A237668, A308546, A326020, A326083, A364349, A365069.
%K A124506 nonn,more
%O A124506 1,3
%A A124506 P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006