A364914 -id:A364914 - cp's OEIS frontend

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 3, 5, 3, 6, 5, 7, 6, 9, 7, 11, 10, 14, 12, 16, 15, 20, 17, 24, 22, 27, 29, 32, 30, 41, 36, 49, 45, 50, 52, 65, 63, 70, 77, 80, 83, 104, 98, 107, 116, 126, 134, 152, 148, 162, 180, 196, 195, 227, 227, 238, 272, 271, 293, 333, 325

Offset: 0

Gus Wiseman, Aug 15 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(16) = 6 through a(22) = 12 strict partitions:
  (16)     (17)     (18)     (19)     (20)      (21)      (22)
  (9,7)    (9,8)    (10,8)   (10,9)   (11,9)    (12,9)    (13,9)
  (10,6)   (10,7)   (11,7)   (11,8)   (12,8)    (13,8)    (14,8)
  (11,5)   (11,6)   (13,5)   (12,7)   (13,7)    (15,6)    (15,7)
  (13,3)   (12,5)   (14,4)   (13,6)   (14,6)    (16,5)    (16,6)
  (7,5,4)  (13,4)   (7,6,5)  (14,5)   (17,3)    (17,4)    (17,5)
           (14,3)   (8,7,3)  (15,4)   (8,7,5)   (19,2)    (18,4)
           (15,2)            (16,3)   (9,6,5)   (11,10)   (19,3)
           (7,6,4)           (17,2)   (9,7,4)   (8,7,6)   (12,10)
                             (8,6,5)  (11,5,4)  (9,7,5)   (9,7,6)
                             (9,6,4)            (10,7,4)  (9,8,5)
                                                (10,8,3)  (7,6,5,4)
                                                (11,6,4)
                                                (11,7,3)

For sums of subsets instead of combinations of partitions we have A151897.
For sums instead of combinations we have A237667, binary A236912.
For subsets instead of partitions we have A326083, complement A364914.
The complement in strict partitions is A364839, non-strict A364913.
A more strict variation is A364915.
The case of all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A116861 and A364916 count linear combinations of strict partitions.
A323092 (ranks A320340) and A120641 count double-free partitions.
A364912 counts linear combinations of partitions of k.
Cf. A007865, A085489, A237113, A275972, A363226, A364272, A364533, A364910, A364911, A365002, A365004.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Table[combs[#[[k]],Delete[#,k]]=={},{k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364350(n):
    if n <= 1: return 1
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

More terms and offset corrected by Martin Fuller, Sep 11 2023

A093971 Number of sum-full subsets of {1,...,n}; subsets A such that there is a solution to x+y=z for x,y,z in A.

Original entry on oeis.org

0, 1, 2, 7, 16, 40, 86, 195, 404, 873, 1795, 3727, 7585, 15537, 31368, 63582, 127933, 257746, 517312, 1038993, 2081696, 4173322, 8355792, 16731799, 33484323, 67014365, 134069494, 268234688, 536562699, 1073326281, 2146849378, 4294117419, 8588623348, 17178130162

Offset: 1

T. D. Noe, Apr 20 2004

nonn

In sumset notation, number of subsets A of {1,...,n} such that the intersection of A and 2A is nonempty.

A variation of binary sum-full sets where parts can be re-used, this sequence counts subsets of {1..n} containing a part equal to the sum of two other (possibly equal) parts. The complement is counted by A007865. The non-binary version is A364914. For non-re-usable parts we have A088809. - Gus Wiseman, Aug 14 2023

			The a(1) = 0 through a(5) = 16 subsets:
  .  {1,2}  {1,2}    {1,2}      {1,2}
            {1,2,3}  {2,4}      {2,4}
                     {1,2,3}    {1,2,3}
                     {1,2,4}    {1,2,4}
                     {1,3,4}    {1,2,5}
                     {2,3,4}    {1,3,4}
                     {1,2,3,4}  {1,4,5}
                                {2,3,4}
                                {2,3,5}
                                {2,4,5}
                                {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}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..88
Eric Weisstein's World of Mathematics, Sum-Free Set

The complement is counted by A007865.
The version without re-usable parts is A088809 (differences A364756), complement A085489 (differences A364755).
The non-binary version is A364914, complement A326083.
The non-binary version w/o re-usable parts is A364534, complement A151897.
The version for partitions is A363225:
- ranks A364348,
- strict A363226,
- non-binary A364839,
- without re-usable parts A237113,
- non-binary without re-usable parts A237668.
The complement for partitions is A364345:
- ranks A364347,
- strict A364346,
- non-binary A364350,
- without re-usable parts A236912,
- non-binary without re-usable parts A237667.
Cf. A000079, A050291, A051026, A103580, A308546, A326080, A364349, A364461, A364533, A364670.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Tuples[#,2]]!={}&]],{n,0,10}] (* Gus Wiseman, Aug 14 2023 *)

a(n) = 2^n - A007865(n).

Terms a(31) and beyond from Fausto A. C. Cariboni, Oct 01 2020

A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 7, 12, 12, 17, 20, 26, 29, 39, 43, 54, 62, 77, 88, 107, 122, 148, 168, 200, 229, 267, 308, 360, 407, 476, 536, 623, 710, 812, 917, 1050, 1190, 1349, 1530, 1733, 1944, 2206, 2483, 2794, 3138, 3524

Offset: 0

Gus Wiseman, Aug 19 2023

nonn

			For y = (4,3,2) we can write 4 = 0*3 + 2*2, so y is counted under a(9).
For y = (11,5,3) we can write 11 = 1*5 + 2*3, so y is counted under a(19).
For y = (17,5,4,3) we can write 17 = 1*3 + 1*4 + 2*5, so y is counted under a(29).
The a(1) = 0 through a(12) = 12 strict partitions (A = 10, B = 11):
  .  .  (21)  (31)  (41)  (42)   (61)   (62)   (63)   (82)    (A1)    (84)
                          (51)   (421)  (71)   (81)   (91)    (542)   (93)
                          (321)         (431)  (432)  (532)   (632)   (A2)
                                        (521)  (531)  (541)   (641)   (B1)
                                               (621)  (631)   (731)   (642)
                                                      (721)   (821)   (651)
                                                      (4321)  (5321)  (732)
                                                                      (741)
                                                                      (831)
                                                                      (921)
                                                                      (5421)
                                                                      (6321)

For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
Non-strict versions are A364913 and the complement of A364915.
For subsets instead of partitions we have A364914, complement A326083.
The case of no all positive coefficients is A365006.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A363226, A365002.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Or@@Table[combs[#[[k]], Delete[#,k]]!={}, {k,Length[#]}]&]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A364839(n):
    if n <= 1: return 0
    alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 0
    for p in partitions(n,k=n-1):
        if max(p.values(),default=0)==1:
            s = set(p)
            if any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
                c += 1
    return c # Chai Wah Wu, Sep 23 2023

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

Original entry on oeis.org

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.

A088314 Cardinality of set of sets of parts of all partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 51, 61, 79, 96, 124, 148, 186, 222, 275, 326, 400, 473, 575, 673, 811, 946, 1132, 1317, 1558, 1813, 2138, 2463, 2893, 3323, 3882, 4461, 5177, 5917, 6847, 7818, 8994, 10251, 11766, 13334, 15281, 17309, 19732, 22307

Offset: 0

Naohiro Nomoto, Nov 05 2003

Number of different values of A007947(m) when A056239(m) is equal to n.
From Gus Wiseman, Sep 11 2023: (Start)
Also the number of finite sets of positive integers that can be linearly combined using all positive coefficients to obtain n. For example, the a(1) = 1 through a(7) = 12 sets are:
  {1}  {1}  {1}    {1}    {1}    {1}      {1}
       {2}  {3}    {2}    {5}    {2}      {7}
            {1,2}  {4}    {1,2}  {3}      {1,2}
                   {1,2}  {1,3}  {6}      {1,3}
                   {1,3}  {1,4}  {1,2}    {1,4}
                          {2,3}  {1,3}    {1,5}
                                 {1,4}    {1,6}
                                 {1,5}    {2,3}
                                 {2,4}    {2,5}
                                 {1,2,3}  {3,4}
                                          {1,2,3}
                                          {1,2,4}
Cf. A365073, A365311, A365312, A365322, A365380.
(End)

			The 7 partitions of 5 and their sets of parts are
[ #]  partition      set of parts
[ 1]  [ 1 1 1 1 1 ]  {1}
[ 2]  [ 2 1 1 1 ]    {1, 2}
[ 3]  [ 2 2 1 ]      {1, 2}  (same as before)
[ 4]  [ 3 1 1 ]      {1, 3}
[ 5]  [ 3 2 ]        {2, 3}
[ 6]  [ 4 1 ]        {1, 4}
[ 7]  [ 5 ]          {5}
so we have a(5) = |{{1}, {1, 2}, {1, 3}, {2, 3}, {1, 4}, {5}}| = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A182410.
The complement in subsets of {1..n-1} is A070880(n) = A365045(n) - 1.
The case of pairs is A365315, see also A365314, A365320, A365321.
A116861 and A364916 count linear combinations of strict partitions.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364914 and A365046 count combination-full subsets.
Cf. A000009, A007865, A088528, A088809, A093971, A326020, A364272, A364534, A365043, A364350.

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

list2set := L -> {op(L)};
a:= N -> list2set(map( list2set, combinat[partition](N) ));
seq(nops(a(n)), n=0..30);
#  Yogy Namara (yogy.namara(AT)gmail.com), Jan 13 2010
b:= proc(n, i) option remember; `if`(n=0, {{}}, `if`(i<1, {},
      {b(n, i-1)[], seq(map(x->{x[],i}, b(n-i*j, i-1))[], j=1..n/i)}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);
# Alois P. Heinz, Aug 09 2012

Table[Length[Union[Map[Union,IntegerPartitions[n]]]],{n,1,30}] (* Geoffrey Critzer, Feb 19 2013 *)
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i < 1, {},
     Union@Flatten@{b[n, i - 1], Table[If[Head[#] == List,
     Append[#, i]]& /@ b[n - i*j, i - 1], {j, 1, n/i}]}]];
a[n_] := Length[b[n, n]];
a /@ Range[0, 40] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)
combp[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,1,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&&combp[n,#]!={}&]], {n,0,15}] (* Gus Wiseman, Sep 11 2023 *)

from sympy.utilities.iterables import partitions
def A088314(n): return len({tuple(sorted(set(p))) for p in partitions(n)}) # Chai Wah Wu, Sep 10 2023

a(n) = 2^(n-1) - A070880(n). - Alois P. Heinz, Feb 08 2019

a(n) = A365042(n) + 1. - Gus Wiseman, Sep 13 2023

More terms and clearer definition from Vladeta Jovovic, Apr 21 2005

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).

A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

Original entry on oeis.org

1, 2, 1, 4, 2, 2, 1, 8, 4, 4, 5, 2, 2, 1, 16, 8, 8, 10, 10, 7, 5, 5, 2, 2, 1, 32, 16, 16, 20, 20, 23, 15, 15, 12, 12, 8, 5, 5, 2, 2, 1, 64, 32, 32, 40, 40, 46, 47, 38, 33, 35, 29, 28, 21, 17, 14, 13, 8, 5, 5, 2, 2, 1, 128, 64, 64, 80, 80, 92, 94, 102, 79, 82, 76, 75, 68, 64, 53, 48, 43, 34, 33, 23, 19, 15, 13, 8, 5, 5, 2, 2, 1

Offset: 0

Gus Wiseman, Sep 08 2023

Row lengths are A000124(n) = 1 + n*(n+1)/2.

			Triangle begins:
   1
   2  1
   4  2  2  1
   8  4  4  5  2  2  1
  16  8  8 10 10  7  5  5  2  2  1
  32 16 16 20 20 23 15 15 12 12  8  5  5  2  2  1
  64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13  8  5  5  2  2  1
Array begins:
     k=0   k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9
-------------------------------------------------------
n=0:  1
n=1:  2     1
n=2:  4     2    2    1
n=3:  8     4    4    5    2    2    1
n=4:  16    8    8    10   10   7    5    5    2    2
n=5:  32    16   16   20   20   23   15   15   12   12
n=6:  64    32   32   40   40   46   47   38   33   35
n=7:  128   64   64   80   80   92   94   102  79   82
n=8:  256   128  128  160  160  184  188  204  207  184
n=9:  512   256  256  320  320  368  376  408  414  440
The T(5,8) = 12 subsets are:
  {3,5}  {1,2,5}  {1,2,3,4}  {1,2,3,4,5}
         {1,3,4}  {1,2,3,5}
         {1,3,5}  {1,2,4,5}
         {2,3,5}  {1,3,4,5}
         {3,4,5}  {2,3,4,5}

Row lengths are A000124 = number of distinct sums of subsets of {1..n}.
Central column/main diagonal is A365376.
A000009 counts sets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365046 counts combination-full subsets, differences of A364914.
Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.

Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],k]&]],{n,0,8},{k,0,n*(n+1)/2}]

A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 10, 12, 20, 27, 39, 51, 74, 95, 130, 169, 225, 288, 378, 479, 617, 778, 990, 1239, 1560, 1938, 2419, 2986, 3696, 4538, 5575, 6810, 8319, 10102, 12274, 14834, 17932, 21587, 25963, 31120, 37275, 44513, 53097, 63181, 75092, 89030, 105460, 124647

Offset: 0

Gus Wiseman, Aug 20 2023

nonn

Includes all non-strict partitions (A047967).

			The a(0) = 0 through a(7) = 12 partitions:
  .  .  (11)  (21)   (22)    (41)     (33)      (61)
              (111)  (31)    (221)    (42)      (322)
                     (211)   (311)    (51)      (331)
                     (1111)  (2111)   (222)     (421)
                             (11111)  (321)     (511)
                                      (411)     (2221)
                                      (2211)    (3211)
                                      (3111)    (4111)
                                      (21111)   (22111)
                                      (111111)  (31111)
                                                (211111)
                                                (1111111)
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(12).
The partition (6,4,3,2) has 6 = 4+2, or 6 = 3+3, or 6 = 2+2+2, or 4 = 2+2, so is counted under a(15).

The strict case is A364839.
For sums instead of combinations we have A364272, binary A364670.
The complement in strict partitions is A364350.
For subsets instead of partitions we have A364914, complement A326083.
Allowing equal parts gives A365068, complement A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A365006 = no strict partitions w/ pos linear combination.
Cf. A085489, A151897, A236912, A237113, A237667, A275972, A364346.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#||Or@@Table[combs[#[[k]],Delete[#,k]]!={},{k,Length[#]}]&]],{n,0,15}]

a(n) + A364915(n) = A000041(n).

A365073 Number of subsets of {1..n} that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

1, 1, 3, 6, 14, 26, 60, 112, 244, 480, 992, 1944, 4048, 7936, 16176, 32320, 65088, 129504, 261248, 520448, 1046208, 2090240, 4186624, 8365696, 16766464, 33503744, 67064064, 134113280, 268347392, 536546816, 1073575936, 2146703360, 4294425600, 8588476416, 17178349568

Offset: 0

Gus Wiseman, Sep 01 2023

nonn

			The subset {2,3,6} has 7 = 2*2 + 1*3 + 0*6 so is counted under a(7).
The a(1) = 1 through a(4) = 14 subsets:
  {1}  {1}    {1}      {1}
       {2}    {3}      {2}
       {1,2}  {1,2}    {4}
              {1,3}    {1,2}
              {2,3}    {1,3}
              {1,2,3}  {1,4}
                       {2,3}
                       {2,4}
                       {3,4}
                       {1,2,3}
                       {1,2,4}
                       {1,3,4}
                       {2,3,4}
                       {1,2,3,4}

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

The case of positive coefficients is A088314.
The case of subsets containing n is A131577.
The binary version is A365314, positive A365315.
The binary complement is A365320, positive A365321.
The positive complement is counted by A365322.
A version for partitions is A365379, strict A365311.
The complement is counted by A365380.
The case of subsets without n is A365542.
A326083 and A124506 appear to count combination-free subsets.
A179822 and A326080 count sum-closed subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A007865, A088809, A093971, A151897, A237668, A308546, A326020, A364534, A364839, A365043, A365381.

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]],combs[n,#]!={}&]],{n,0,5}]

a(n)={
  my(comb(k,b)=while(b>>k, b=bitor(b, b>>k); k*=2); b);
  my(recurse(k,b)=
    if(bittest(b,0), 2^(n+1-k),
    if(2*k>n, 2^(n+1-k) - 2^sum(j=k, n, !bittest(b,j)),
    self()(k+1, b) + self()(k+1, comb(k,b)) )));
  recurse(1, 1<Andrew Howroyd, Sep 04 2023

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

A365380 Number of subsets of {1..n} that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 16, 12, 32, 32, 104, 48, 256, 208, 448, 448, 1568, 896, 3840, 2368, 6912, 7680, 22912, 10752, 50688, 44800, 104448, 88064, 324096, 165888, 780288, 541696, 1458176, 1519616, 4044800, 2220032, 10838016, 8744960, 20250624, 16433152, 62267392, 34865152

Offset: 1

Gus Wiseman, Sep 04 2023

nonn

			The set {4,5,6} cannot be linearly combined to obtain 7 so is counted under a(7), but we have 8 = 2*4 + 0*5 + 0*6, so it is not counted under a(8).
The a(1) = 1 through a(8) = 12 subsets:
  {}  {}  {}   {}   {}     {}     {}       {}
          {2}  {3}  {2}    {4}    {2}      {3}
                    {3}    {5}    {3}      {5}
                    {4}    {4,5}  {4}      {6}
                    {2,4}         {5}      {7}
                    {3,4}         {6}      {3,6}
                                  {2,4}    {3,7}
                                  {2,6}    {5,6}
                                  {3,5}    {5,7}
                                  {3,6}    {6,7}
                                  {4,5}    {3,6,7}
                                  {4,6}    {5,6,7}
                                  {5,6}
                                  {2,4,6}
                                  {3,5,6}
                                  {4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 1..100
Steven R. Finch, Monoids of natural numbers, March 17, 2009.

The complement is counted by A365073, without n A365542.
The binary complement is A365314, positive A365315.
The binary case is A365320, positive A365321.
For positive coefficients we have A365322, complement A088314.
A124506 appears to count combination-free subsets, differences of A326083.
A179822 counts sum-closed subsets, first differences of A326080.
A288728 counts binary sum-free subsets, first differences of A007865.
A365046 counts combination-full subsets, first differences of A364914.
A365071 counts sum-free subsets, first differences of A151897.
Cf. A050291, A085489, A088528, A088809, A093971, A326020, A364350, A364534, A365043, A365045.

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-1]],combs[n,#]=={}&]],{n,5}]

a(n) = 2^n - A365073(n).

Terms a(12) and beyond from Andrew Howroyd, Sep 04 2023

cp's OEIS Frontend

A364350 Number of strict integer partitions of n such that no part can be written as a nonnegative linear combination of the others.

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A093971 Number of sum-full subsets of {1,...,n}; subsets A such that there is a solution to x+y=z for x,y,z in A.

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A364839 Number of strict integer partitions of n such that some part can be written as a nonnegative linear combination of the others.

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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.

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A088314 Cardinality of set of sets of parts of all partitions of 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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A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.

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A364913 Number of integer partitions of n having a part that can be written as a nonnegative linear combination of the other (possibly equal) parts.

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