A158206 -id:A158206 - cp's OEIS frontend

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

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

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)

			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.

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).

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.

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.

A158278 Number of symmetric numerical semigroups with Frobenius number 2n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2n-1.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 8, 7, 15, 20, 18, 36, 44, 45, 83, 109, 101, 174, 246, 227

Offset: 1

Steven Finch, Mar 15 2009

			a(3)=2: the only 2 symmetric semigroups with Frobenius number 5=2*3-1 are generated by {3, 4} and {2, 7}.

CombOS - Combinatorial Object Server, Generate numerical semigroups
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).

Cf. A124506, A158206.

a(n) = A158206(2*n-1).

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2n.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 6, 7, 7, 11, 20, 14, 35, 37, 36, 70, 106, 77, 182

Offset: 1

Steven Finch, Mar 15 2009

			a(3)=1: the unique pseudo-symmetric semigroup with Frobenius number 6=2*3 is generated by {4, 5, 7}.

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).

Cf. A124506, A158206.

a(n) = A158206(2*n).

A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 5, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 7, 6, 1, 1, 4, 2, 1, 8, 9, 2, 1, 5, 4, 1, 1, 7, 8, 2, 1, 8, 9, 2, 1, 10, 17, 7, 1, 1, 5, 6, 2, 1, 10, 19, 12, 2, 1, 10, 16, 7, 1, 1, 10, 21, 11, 2, 1, 9, 16, 9, 2, 1, 13, 34, 26, 8, 1, 1, 8, 15, 10, 2, 1, 14, 41, 37, 14, 2

Offset: 1

Christopher O'Neill, Sep 24 2018

The length of each row is floor((n+1)/2) - floor(n/3).
Summing rows yields A158206.
The expected number of minimal generators of a randomly selected numerical semigroup S(M,p) equals Sum_{n=1..M} ( p * (1 - p)^(floor(n/2)) * Product_{k>=0} T(n,k)*p^k ).

			T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.
When written in rows:
  1
  1
  1
  1
  1,  1
  1
  1,  2
  1,  1
  1,  2
  1,  2
  1,  4,  1
  1,  1
  1,  5,  2
  1,  4,  1
  1,  4,  2
  1,  4,  2
  1,  7,  6,  1
  1,  4,  2
  1,  8,  9,  2
  1,  5,  4,  1
  1,  7,  8,  2
  1,  8,  9,  2
  1, 10, 17,  7,  1
  1,  5,  6,  2
  1, 10, 19, 12,  2
  1, 10, 16,  7,  1
  1, 10, 21, 11,  2
  1,  9, 16,  9,  2
  1, 13, 34, 26,  8,  1
  1,  8, 15, 10,  2

J. De Loera, C. O'Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, arXiv:1710.00979 [math.AC], 2017.
C. Leng and C. O'Neill, A sequence of quasipolynomials arising from random numerical semigroups, arXiv:1809.09915 [math.CO], 2018.
C. O'Neill, The first 90 rows formatted as a triangle

Cf. A008615, A158206.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

A158278 Number of symmetric numerical semigroups with Frobenius number 2n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

A158278 Number of symmetric numerical semigroups with Frobenius number 2*n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2*n-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2*n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2*n.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A158278 Number of symmetric numerical semigroups with Frobenius number 2n-1; that is, symmetric numerical semigroups for which the largest integer not belonging to them is 2n-1.

A158279 Number of pseudo-symmetric numerical semigroups with Frobenius number 2n; that is, pseudo-symmetric numerical semigroups for which the largest integer not belonging to them is 2n.