A124506 - OEIS

%I #29 Sep 02 2023 11:28:45

%S 1,1,2,2,5,4,11,10,21,22,51,40,106,103,200,205,465,405,961,900,1828,

%T 1913,4096,3578,8273,8175,16132,16267,34903,31822,70854,68681,137391,

%U 140661,292081,270258,591443,582453,1156012

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

%C From _Gus Wiseman_, Aug 28 2023: (Start)

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

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

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

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

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

%C                                             {6,7}      {3,7,8}

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

%C                                             {4,5,7}    {5,7,8}

%C                                             {4,6,7}    {6,7,8}

%C                                             {5,6,7}    {5,6,7,8}

%C                                             {4,5,6,7}

%C 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 (End)

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Monoids of natural numbers</a>

%H S. R. Finch, <a href="/A066062/a066062.pdf">Monoids of natural numbers</a>, March 17, 2009. [Cached copy, with permission of the author]

%H 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 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 a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.

%e a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.

%e 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 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 (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 Cf. A158206. [From _Steven Finch_, Mar 13 2009]

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

%Y A364350 counts combination-free partitions, complement A364839.

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

%K nonn,more

%O 1,3

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