A365069 -id:A365069 - 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.

A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 1, 2, 6, 11, 28, 53, 118, 235, 490, 973, 2008, 3990, 8089, 16184, 32563, 65071, 130667, 261183, 523388, 1046748, 2095239, 4190208, 8385030, 16768943, 33546257, 67092732, 134201461, 268400553, 536839090, 1073670970, 2147414967, 4294829905, 8589793931

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Includes all subsets containing both 1 and n.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}      {1,5}
               {1,2,3}  {2,4}      {1,2,5}
                        {1,2,4}    {1,3,5}
                        {1,3,4}    {1,4,5}
                        {2,3,4}    {2,3,5}
                        {1,2,3,4}  {2,4,5}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {2,3,4,5}
                                   {1,2,3,4,5}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A364914.
The positive version is A365042, first differences of A365043.
The positive complement is counted by A365045, first differences of A365044.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A364350 counts combination-free strict partitions, complement A364839.
A085489 and A364755 count subsets without the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combs[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n+1) = 2^n - A124506(n).

A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 3, 7, 12, 21, 32, 49, 70, 99, 135, 185, 245, 323, 418, 541, 688, 873, 1094, 1368, 1693, 2092, 2564, 3138, 3810, 4620, 5565, 6696, 8012, 9569, 11381, 13518, 15980, 18872, 22194, 26075, 30535, 35711, 41627, 48473, 56290, 65283, 75533, 87298, 100631, 115911, 133219

Offset: 0

Gus Wiseman, Aug 25 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} such that some element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,4,9} has 9 = 3*3 + 0*4, but this is not strictly positive, so S is not counted under a(9).
The subset S = {3,4,10} has 10 = 2*3 + 1*4, so S is counted under a(10).
The a(0) = 0 through a(5) = 12 subsets:
  .  .  {1,2}  {1,2}    {1,2}    {1,2}
               {1,3}    {1,3}    {1,3}
               {1,2,3}  {1,4}    {1,4}
                        {2,4}    {1,5}
                        {1,2,3}  {2,4}
                        {1,2,4}  {1,2,3}
                        {1,3,4}  {1,2,4}
                                 {1,2,5}
                                 {1,3,4}
                                 {1,3,5}
                                 {1,4,5}
                                 {2,3,5}

Bert Dobbelaere, Table of n, a(n) for n = 0..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary complement is A007865, first differences A288728.
The binary version is A093971, first differences A365070.
The nonnegative complement is A326083, first differences A124506.
The nonnegative version is A364914, first differences A365046.
First differences are A365042.
The complement is counted by A365044, first differences A365045.
Without re-usable parts we have A364534, first differences A365069.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Rest[Subsets[Range[n]]],combp[Last[#],Union[Most[#]]]!={}&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365043(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365044(n).

a(15)-a(35) from Chai Wah Wu, Nov 20 2023

More terms from Bert Dobbelaere, Apr 28 2025

A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

Original entry on oeis.org

0, 1, 1, 2, 4, 11, 23, 53, 111, 235, 483, 988, 1998, 4036, 8114, 16289, 32645, 65389, 130887, 261923, 524014, 1048251, 2096753, 4193832, 8388034, 16776544, 33553622, 67107919, 134216597, 268434140, 536869355, 1073740012, 2147481511, 4294964834, 8589931700

Offset: 0

Gus Wiseman, Aug 24 2023

nonn

Also subsets of {1..n} containing n whose greatest element cannot be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is not counted under a(10).
The a(0) = 0 through a(5) = 11 subsets:
  .  {1}  {2}  {3}    {4}        {5}
               {2,3}  {3,4}      {2,5}
                      {2,3,4}    {3,5}
                      {1,2,3,4}  {4,5}
                                 {2,4,5}
                                 {3,4,5}
                                 {1,2,3,5}
                                 {1,2,4,5}
                                 {1,3,4,5}
                                 {2,3,4,5}
                                 {1,2,3,4,5}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The nonempty case is A070880.
The nonnegative version is A124506, first differences of A326083.
The binary version is A288728, first differences of A007865.
A subclass is A341507.
The complement is counted by A365042, first differences of A365043.
First differences of A365044.
The nonnegative complement is A365046, first differences of A364914.
The binary complement is A365070, first differences of A093971.
Without re-usable parts we have A365071, first differences of A151897.
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&And@@Table[combp[#[[k]],Union[Delete[#,k]]]=={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A070880(n) + 1 for n > 0.

A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

Original entry on oeis.org

1, 2, 3, 5, 9, 20, 43, 96, 207, 442, 925, 1913, 3911, 7947, 16061, 32350, 64995, 130384, 261271, 523194, 1047208, 2095459, 4192212, 8386044, 16774078, 33550622, 67104244, 134212163, 268428760, 536862900, 1073732255, 2147472267, 4294953778, 8589918612, 17179850312

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

Sets of this type may be called "positive combination-free".

Also subsets of {1..n} such that no element can be written as a (strictly) positive linear combination of the others.

			The subset S = {3,5,6,8} has 6 = 2*3 + 0*5 + 0*8 and 8 = 1*3 + 1*5 + 0*6 but neither of these is strictly positive, so S is counted under a(8).
The a(0) = 1 through a(5) = 20 subsets:
  {}  {}   {}   {}     {}         {}
      {1}  {1}  {1}    {1}        {1}
           {2}  {2}    {2}        {2}
                {3}    {3}        {3}
                {2,3}  {4}        {4}
                       {2,3}      {5}
                       {3,4}      {2,3}
                       {2,3,4}    {2,5}
                       {1,2,3,4}  {3,4}
                                  {3,5}
                                  {4,5}
                                  {2,3,4}
                                  {2,4,5}
                                  {3,4,5}
                                  {1,2,3,4}
                                  {1,2,3,5}
                                  {1,2,4,5}
                                  {1,3,4,5}
                                  {2,3,4,5}
                                  {1,2,3,4,5}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The binary version is A007865, first differences A288728.
The binary complement is A093971, first differences A365070.
Without re-usable parts we have A151897, first differences A365071.
The nonnegative version is A326083, first differences A124506.
A subclass is A341507.
The nonnegative complement is A364914, first differences A365046.
The complement is counted by A365043, first differences A365042.
First differences are A365045.
A085489 and A364755 count subsets w/o the sum of two distinct elements.
A088809 and A364756 count subsets with the sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A006951, A237113, A237668, A308546, A324736, A326020, A326080, A364272, A364349, A364534, A365069.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],And@@Table[combp[Last[#],Union[Most[#]]]=={},{k,Length[#]}]&]],{n,0,10}]

from itertools import combinations
from sympy.utilities.iterables import partitions
def A365044(n):
    mlist = tuple({tuple(sorted(p.keys())) for p in partitions(m,k=m-1)} for m in range(1,n+1))
    return n+1+sum(1 for k in range(2,n+1) for w in combinations(range(1,n+1),k) if w[:-1] not in mlist[w[-1]-1]) # Chai Wah Wu, Nov 20 2023

a(n) = 2^n - A365043(n).

a(15)-a(34) from Chai Wah Wu, Nov 20 2023

A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 9, 11, 17, 21, 29, 36, 50, 60, 78, 95, 123, 147, 185, 221, 274, 325, 399, 472, 574, 672, 810, 945, 1131, 1316, 1557, 1812, 2137, 2462, 2892, 3322, 3881, 4460, 5176, 5916, 6846, 7817, 8993, 10250, 11765, 13333, 15280, 17308, 19731, 22306

Offset: 0

Gus Wiseman, Aug 23 2023

nonn

Sets of this type may be called "positive combination-full".

Also subsets of {1..n} containing n whose greatest element can be written as a positive linear combination of the others.

			The subset {3,4,10} has 10 = 2*3 + 1*4 so is counted under a(10).
The a(0) = 0 through a(7) = 11 subsets:
  .  .  {1,2}  {1,3}    {1,4}    {1,5}    {1,6}      {1,7}
               {1,2,3}  {2,4}    {1,2,5}  {2,6}      {1,2,7}
                        {1,2,4}  {1,3,5}  {3,6}      {1,3,7}
                        {1,3,4}  {1,4,5}  {1,2,6}    {1,4,7}
                                 {2,3,5}  {1,3,6}    {1,5,7}
                                          {1,4,6}    {1,6,7}
                                          {1,5,6}    {2,3,7}
                                          {2,4,6}    {2,5,7}
                                          {1,2,3,6}  {3,4,7}
                                                     {1,2,3,7}
                                                     {1,2,4,7}

Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The nonnegative complement is A124506, first differences of A326083.
The binary complement is A288728, first differences of A007865.
First differences of A365043.
The complement is counted by A365045, first differences of A365044.
The nonnegative version is A365046, first differences of A364914.
Without re-usable parts we have A365069, first differences of A364534.
The binary version is A365070, first differences of A093971.
A085489 and A364755 count subsets with no sum of two distinct elements.
A088314 counts sets that can be linearly combined to obtain n.
A088809 and A364756 count subsets with some sum of two distinct elements.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
Cf. A151897, A237113, A237668, A308546, A324736, A326020, A326080, A364272.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Or@@Table[combp[#[[k]],Union[Delete[#,k]]]!={},{k,Length[#]}]&]],{n,0,10}]

a(n) = A088314(n) - 1.

A365071 Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 15, 23, 40, 55, 94, 132, 210, 298, 476, 644, 1038, 1406, 2149, 2965, 4584, 6077, 9426, 12648, 19067, 25739, 38958, 51514, 78459, 104265, 155436, 208329, 312791, 411886, 620780, 823785, 1224414, 1631815, 2437015, 3217077, 4822991

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

The complement is counted by A365069. The binary version is A364755, complement A364756. For re-usable parts we have A288728, complement A365070.

			The subset {1,3,4,6} has 4 = 1 + 3 so is not counted under a(6).
The subset {2,3,4,5,6} has 6 = 2 + 4 and 4 = 1 + 3 so is not counted under a(6).
The a(0) = 0 through a(6) = 15 subsets:
  .  {1}  {2}    {3}    {4}      {5}      {6}
          {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
                 {2,3}  {2,4}    {2,5}    {2,6}
                        {3,4}    {3,5}    {3,6}
                        {1,2,4}  {4,5}    {4,6}
                        {2,3,4}  {1,2,5}  {5,6}
                                 {1,3,5}  {1,2,6}
                                 {2,4,5}  {1,3,6}
                                 {3,4,5}  {1,4,6}
                                          {2,3,6}
                                          {2,5,6}
                                          {3,4,6}
                                          {3,5,6}
                                          {4,5,6}
                                          {3,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..85
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

First differences of A151897.
The version with re-usable parts is A288728 first differences of A007865.
The binary version is A364755, first differences of A085489.
The binary complement is A364756, first differences of A088809.
The complement is counted by A365069, first differences of A364534.
The complement w/ re-usable parts is A365070, first differences of A093971.
A108917 counts knapsack partitions, strict A275972.
A124506 counts combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
Partitions: A236912, A237113, A237668, A364532, A364272, A364349, A364913.
Cf. A050291, A095944, A103580, A324741, A326080, A326117, A341507, A364533.

Table[Length[Select[Subsets[Range[n]], MemberQ[#,n]&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]=={}&]], {n,0,10}]

a(n) + A365069(n) = 2^(n-1).

First differences of A151897.

a(14) onwards added (using A151897) by Andrew Howroyd, Jan 13 2024

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.

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A365046 Number of subsets of {1..n} containing n such that some element can be written as a nonnegative linear combination of the others.

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A365043 Number of subsets of {1..n} whose greatest element can be written as a (strictly) positive linear combination of the others.

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A365045 Number of subsets of {1..n} containing n such that no element can be written as a positive linear combination of the others.

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A365044 Number of subsets of {1..n} whose greatest element cannot be written as a (strictly) positive linear combination of the others.

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A365042 Number of subsets of {1..n} containing n such that some element can be written as a positive linear combination of the others.

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A365071 Number of subsets of {1..n} containing n such that no element is a sum of distinct other elements. A variation of non-binary sum-free subsets without re-usable elements.

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