A364916 -id:A364916 - cp's OEIS frontend

A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

0, 0, 0, 1, 2, 7, 17, 41, 88, 201, 418, 892, 1838, 3798, 7716, 15740

Offset: 0

Gus Wiseman, Aug 26 2023

nonn

The complement is counted by A365071. The binary case is A364756. Allowing elements to be re-used gives A365070. A version for partitions (but not requiring n) is A237668.

			The subset {2,4,6} has 6 = 4 + 2 so is counted under a(6).
The subset {1,2,4,7} has 7 = 4 + 2 + 1 so is counted under a(7).
The subset {1,4,5,8} has 5 = 4 + 1 so is counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                    {1,2,3,4}  {2,3,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

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

The complement w/ re-usable parts is A288728, first differences of A007865.
First differences of A364534.
The binary complement is A364755, first differences of A085489.
The binary version is A364756, first differences of A088809.
The version with re-usable parts is A365070, first differences of A093971.
The complement is counted by A365071, first differences of A151897.
A124506 counts nonnegative combination-free subsets, differences of A326083.
A365046 counts nonnegative combination-full subsets, differences of A364914.
For partitions: A108917, A236912, A237113, A237668, A364532, A364913.
Strict partitions: A116861, A364272, A364349, A364350, A364839, A364916.
Cf. A050291, A326080, A363226, A364346, A364348, A364670.

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

a(n) = 2^(n-1) - A365070(n).

First differences of A364534.

A364906 Number of ways to write A056239(n) as a nonnegative linear combination of the multiset of prime indices of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 1, 10, 3, 2, 1, 9, 1, 2, 1, 35, 1, 6, 1, 9, 2, 2, 1, 34, 3, 2, 10, 10, 1, 7, 1, 126, 1, 2, 1, 30, 1, 2, 2, 39, 1, 6, 1, 11, 3, 2, 1, 130, 3, 6, 1, 12, 1, 20, 1, 46, 2, 2, 1, 31, 1, 2, 9, 462, 2, 7, 1, 13, 1, 6, 1, 120, 1, 2, 4, 14, 1, 7, 1

Offset: 1

Gus Wiseman, Aug 22 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
A way of writing n as a (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).
Conjecture: Positions of 1's are numbers whose distinct divisors all have different GCDs of prime indices, listed by A319319, counted by A319318.

			The a(2) = 1 through a(10) = 2 ways:
  1*1  1*2  0*1+2*1  1*3  1*1+1*2  1*4  0*1+0*1+3*1  0*2+2*2  1*1+1*3
            1*1+1*1       3*1+0*2       0*1+1*1+2*1  1*2+1*2  4*1+0*3
            2*1+0*1                     0*1+2*1+1*1  2*2+0*2
                                        0*1+3*1+0*1
                                        1*1+0*1+2*1
                                        1*1+1*1+1*1
                                        1*1+2*1+0*1
                                        2*1+0*1+1*1
                                        2*1+1*1+0*1
                                        3*1+0*1+0*1

The case with no zero coefficients is A000012.
Positions of 1's appear to be A319319.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, sum A056239.
A364910 counts nonnegative linear combinations of strict partitions.
Cf. A116861, A275972, A320340, A364350, A364839, A364912, A364914, A364916, A365002, A365003, A365004.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[combs[Total[prix[n]],prix[n]]],{n,100}]

A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 7, 10, 16, 23, 34, 44, 67, 85, 119, 157, 210, 268, 360, 453, 592, 748, 956, 1195, 1520, 1883, 2365, 2920, 3628, 4451, 5494, 6702, 8211, 9976, 12147, 14666, 17776, 21389, 25774, 30887, 37035, 44224, 52819, 62836, 74753, 88614, 105062, 124160

Offset: 0

Gus Wiseman, Aug 27 2023

nonn

These may be called "non-binary nonnegative combination-full" partitions.

Does not necessarily include all non-strict partitions (A047967).

			The partition (5,4,3,3) has no part that can be written as a nonnegative linear combination of the others, so is not counted under a(15).
The partition (6,4,3,2) has 6 = 1*2 + 1*4, so is counted under a(15). The combinations 6 = 2*3 = 3*2 and 4 = 2*2 can also be used.
The a(3) = 1 through a(8) = 16 partitions:
  (21)  (31)   (41)    (42)     (61)      (62)
        (211)  (221)   (51)     (331)     (71)
               (311)   (321)    (421)     (422)
               (2111)  (411)    (511)     (431)
                       (2211)   (2221)    (521)
                       (3111)   (3211)    (611)
                       (21111)  (4111)    (3221)
                                (22111)   (3311)
                                (31111)   (4211)
                                (211111)  (5111)
                                          (22211)
                                          (32111)
                                          (41111)
                                          (221111)
                                          (311111)
                                          (2111111)

The complement for sums instead of combinations is A237667, binary A236912.
For sums instead of combinations we have A237668, binary A237113.
The strict case is A364839, complement A364350.
Allowing equal parts in the combination gives A364913.
For subsets instead of partitions we have A364914, complement A326083.
The complement is A364915.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A323092 counts double-free partitions, ranks A320340.
A364912 counts linear combinations of partitions of k.
Cf. A108917, A151897, A364272, A364910, A364911, A365006.

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], Function[ptn,Or@@Table[combs[ptn[[k]], DeleteCases[ptn,ptn[[k]]]]!={}, {k,Length[ptn]}]]]],{n,0,5}]

from sympy.utilities.iterables import partitions
def A365068(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):
        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 20 2023

a(31)-a(47) from Chai Wah Wu, Sep 20 2023

A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 2, 4, 12, 8, 20, 11, 14, 26, 43, 19, 38, 53, 51, 48, 101, 48, 124, 96, 121, 159, 134, 103, 241, 261, 244, 175, 401, 229, 488, 358, 328

Offset: 0

Gus Wiseman, Sep 08 2023

			The a(11) = 2 through a(18) = 8 partitions:
  (5,4)  .  (6,5)  (6,5)   (7,6)  (7,5)   (7,4)     (7,5)
  (7,3)     (7,4)  (8,5)   (9,4)  (7,6)   (7,6)     (8,7)
            (7,5)  (9,4)          (9,5)   (8,5)     (10,7)
            (8,3)  (10,3)         (11,3)  (8,7)     (11,4)
                                          (9,5)     (11,5)
                                          (9,7)     (12,5)
                                          (10,3)    (13,4)
                                          (11,4)    (7,5,5)
                                          (11,5)
                                          (13,3)
                                          (7,4,4)
                                          (10,3,3)

Relatively prime partitions are counted by A000837, ranks A289509.
This is the relatively prime case of A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A007359, A289508, A364345, A365073, A365312, A365379, A365380, A365383.

combsu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],GCD@@#==1&&combsu[n,#]=={}&]],{n,0,20}]

from math import gcd
from sympy.utilities.iterables import partitions
def A365382(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n) for b in partitions(m) if gcd(*b.keys()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

a(21)-a(45) from Chai Wah Wu, Sep 13 2023

A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

Original entry on oeis.org

1, 1, 4, 15, 70, 314, 1542, 7428, 36860, 182911, 917188, 4612480, 23323662, 118273428, 601762636, 3069070533, 15689123386, 80356953555, 412300910566, 2118715503962, 10902791722490, 56175374185014, 289766946825180, 1496239506613985, 7733302967423382

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

A way of writing n as a (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(3) = 15 ways to write 3 as a nonnegative linear combination of an integer composition of 3:
  1*3  0*2+3*1  1*1+1*2  0*1+0*1+3*1
       1*2+1*1  3*1+0*2  0*1+1*1+2*1
                         0*1+2*1+1*1
                         0*1+3*1+0*1
                         1*1+0*1+2*1
                         1*1+1*1+1*1
                         1*1+2*1+0*1
                         2*1+0*1+1*1
                         2*1+1*1+0*1
                         3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A011782.
The version for partitions is A364907, strict A364910.
The strict case is A364909.
A000041 counts integer partitions, strict A000009.
A011782 counts compositions, strict A032020.
A097805 counts compositions by length, strict A072574.
A116861 = positive linear combinations of strict ptns of k, reverse A364916.
A365067 = nonnegative linear combinations of strict partitions of k.
A364912 = positive linear combinations of partitions of k.
A364916 = positive linear combinations of strict partitions of k.
Cf. A364350, A364839, A364906, A364911, A364914, A365002, A365004.

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
      add(add(b(n-i, m-i*j), j=0..m/i), i=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,Join@@Permutations /@ IntegerPartitions[n]}]],{n,0,5}]

a(8)-a(24) from Alois P. Heinz, Jan 28 2024

A364909 Number of ways to write n as a nonnegative linear combination of a strict integer composition of n.

Original entry on oeis.org

1, 1, 1, 5, 5, 7, 51, 45, 89, 109, 709, 733, 1495, 1935, 3119, 13785, 16611, 29035, 44611, 68733, 95193, 372897, 435007, 781345, 1177181, 1866659, 2600537, 3906561, 12052631, 14610799, 25407653, 37652265, 59943351, 84060993, 128112805, 172172117, 480353257, 578740011

Offset: 0

Gus Wiseman, Aug 18 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(0) = 1 through a(5) = 7 ways:
  .  1*1  1*2  1*3      1*4      1*5
               0*2+3*1  0*3+4*1  0*4+5*1
               1*1+1*2  1*1+1*3  1*1+1*4
               1*2+1*1  1*3+1*1  1*2+1*3
               3*1+0*2  4*1+0*3  1*3+1*2
                                 1*4+1*1
                                 5*1+0*4

The case with no zero coefficients is A032020.
The version for partitions is A364907, strict A364910(n) = A364916(n,n).
The non-strict version is A364908.
A000041 counts integer partitions, strict A000009.
A011782 counts compositions, strict A032020.
A008284 counts partitions by length, strict A008289.
A097805 counts compositions by length, strict A072574.
Cf. A116861, A364350, A364839, A364906, A364912, A364914, 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[Join@@Table[combs[n,ptn],{ptn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,5}]

from math import factorial
from sympy.utilities.iterables import partitions
def A364909(n):
    if n == 0: return 1
    aset = tuple(set(p) for p in partitions(n) if max(p.values(),default=0)==1)
    return sum(factorial(len(t)) for p in partitions(n) for t in aset if set(p).issubset(t)) # Chai Wah Wu, Sep 21 2023

a(18)-a(37) from Chai Wah Wu, Sep 21 2023

A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

Original entry on oeis.org

1, 4, 9, 25, 36, 48, 49, 100, 121, 160, 169, 196, 225, 289, 361, 441, 448, 484, 529, 567, 676, 750, 810, 841, 900, 961, 1080, 1089, 1156, 1200, 1225, 1369, 1408, 1440, 1444, 1521, 1681, 1764, 1849, 1920, 2116, 2209, 2268, 2352, 2601, 2809, 3024, 3025, 3159

Offset: 1

Gus Wiseman, Aug 23 2023

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 750 are {1,2,3,3,3}, with sum 12, while the distinct prime indices {1,2,3} have sum 6, so 750 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    25: {3,3}
    36: {1,1,2,2}
    48: {1,1,1,1,2}
    49: {4,4}
   100: {1,1,3,3}
   121: {5,5}
   160: {1,1,1,1,1,3}
   169: {6,6}
   196: {1,1,4,4}
   225: {2,2,3,3}
   289: {7,7}
   361: {8,8}
   441: {2,2,4,4}
   448: {1,1,1,1,1,1,4}

The LHS is A056239 (sum of prime indices).
The RHS is twice A066328.
Partitions of this type are counted by A364910.
A000041 counts integer partitions, strict A000009.
A001222 counts prime indices, distinct A001221.
A112798 lists prime indices, distinct A304038.
A116861 counts partitions by sum and sum of distinct parts.
A323092 counts double-free partitions, ranks A320340.
Cf. A364350, A364839, A364906, A364907, A364911, A364916.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Total[prix[#]]==2*Total[Union[prix[#]]]&]

A056239(a(n)) = 2*A066328(a(n)).

A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 17, 15, 31, 34, 53, 65, 109, 117, 196, 224, 328, 405, 586, 673, 968, 1163, 1555, 1889, 2531, 2986, 3969, 4744, 6073, 7333, 9317, 11053, 14011, 16710, 20702, 24714, 30549, 36127, 44413, 52561, 63786, 75583, 91377, 107436, 129463

Offset: 0

Gus Wiseman, Aug 31 2023

nonn

We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (1111)  (11111)  (222)     (52)       (53)
                                     (111111)  (322)      (332)
                                               (1111111)  (2222)
                                                          (11111111)
The a(11) = 17 partitions:
  (11)  (9,2)  (7,2,2)  (5,3,2,1)  (4,3,2,1,1)  (1,1,1,1,1,1,1,1,1,1,1)
        (8,3)  (6,3,2)  (5,2,2,2)  (3,2,2,2,2)
        (7,4)  (5,4,2)  (4,3,2,2)
        (6,5)  (5,3,3)  (3,3,3,2)
               (4,4,3)

The nonnegative version is A364915, strict A364350.
The strict case is A365006.
For subsets instead of partitions we have A365044, complement A365043.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A237667 counts sum-free partitions, binary A236912.
A364912 counts positive linear combinations of partitions.
A365068 counts combination-full partitions, strict A364839.
Cf. A085489, A108917, A151897, A325862, A364272, A364910, A364911, A364913.

combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Union/@IntegerPartitions[n], Function[ptn,!Or@@Table[combp[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]],{n,0,15}]

from sympy.utilities.iterables import partitions
def A365072(n):
    if n <= 1: return 1
    alist = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)]
    c = 1
    for p in partitions(n,k=n-1):
        s = set(p)
        for q in s:
            if tuple(sorted(s-{q})) in alist[q]:
                break
        else:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

a(31)-a(49) from Chai Wah Wu, Sep 20 2023

A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 9, 7, 15, 16, 29, 23, 47, 43, 74, 65, 114, 100, 174, 153, 257, 228, 368, 312, 530, 454, 736, 645, 1025, 902, 1402, 1184, 1909, 1626, 2618, 2184, 3412, 2895, 4551, 3887, 5966, 5055, 7796, 6509, 10244, 8462, 13060, 10881, 16834, 14021, 21471

Offset: 1

Gus Wiseman, Sep 12 2023

nonn

			The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
The a(3) = 1 through a(10) = 16 partitions:
  (2)  (3)  (2)    (4)    (2)      (3)    (2)        (3)
            (3)    (5)    (3)      (5)    (4)        (4)
            (4)    (3,2)  (4)      (6)    (5)        (6)
            (2,2)         (5)      (7)    (6)        (7)
                          (6)      (3,3)  (7)        (8)
                          (2,2)    (4,3)  (8)        (9)
                          (3,3)    (5,2)  (2,2)      (3,3)
                          (4,2)           (4,2)      (4,4)
                          (2,2,2)         (4,3)      (5,2)
                                          (4,4)      (5,3)
                                          (5,3)      (5,4)
                                          (6,2)      (6,3)
                                          (2,2,2)    (7,2)
                                          (4,2,2)    (3,3,3)
                                          (2,2,2,2)  (4,3,2)
                                                     (5,2,2)

Chai Wah Wu, Table of n, a(n) for n = 1..95

Complement for subsets: A088314 or A365042, nonnegative A365073 or A365542.
For strict partitions we have A088528, nonnegative coefficients A365312.
For length-2 subsets we have A365321 (we use n instead of n-1).
For subsets we have A365322 or A365045, nonnegative coefficients A365380.
For nonnegative coefficients we have A365378, complement A365379.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

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@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]

from sympy.utilities.iterables import partitions
def A365323(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # Chai Wah Wu, Sep 12 2023

a(21)-a(51) from Chai Wah Wu, Sep 12 2023

A365383 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53

Offset: 0

Gus Wiseman, Sep 08 2023

Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
  22, 15, 19, 17, 20, 17, 20, 16,
  22, 17, 20, 17, 21, 17, 20, 17,
  22, 17, 20, 17, 21, 17, 20, 17, ...

			Triangle begins:
   1
   2   1
   3   2   2
   5   3   4   3
   7   5   6   6   6
  11   7   9   8   9   7
  15  11  13  13  14  13  14
  22  15  19  17  20  17  20  16
  30  22  26  26  27  26  28  26  27
  42  30  37  34  39  33  40  34  39  34
  56  42  50  49  52  50  54  51  54  53  53
  77  56  68  64  71  63  73  63  71  65  70  62
 101  77  91  89  95  90  97  93  97  97  98  94  99
 135 101 122 115 127 115 130 114 131 119 130 117 132 116
 176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
  (6)       (51)      (51)      (51)      (51)      (51)
  (51)      (411)     (42)      (411)     (42)      (411)
  (42)      (321)     (411)     (33)      (411)     (321)
  (411)     (3111)    (321)     (321)     (321)     (3111)
  (33)      (2211)    (3111)    (3111)    (3111)    (2211)
  (321)     (21111)   (222)     (2211)    (222)     (21111)
  (3111)    (111111)  (2211)    (21111)   (2211)    (111111)
  (222)               (21111)   (111111)  (21111)
  (2211)              (111111)            (111111)
  (21111)
  (111111)

Column k = 0 is A000041, strict A000009.
The version for subsets is A365381, main diagonal A365376.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A088314, A088528, A237668, A363225, A364345, A365073, A365311, A365320, A365378, A365382.

combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]

cp's OEIS Frontend

A365069 Number of subsets of {1..n} containing n and some element equal to the sum of two or more distinct other elements. A variation of non-binary sum-full subsets without re-usable elements.

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A364906 Number of ways to write A056239(n) as a nonnegative linear combination of the multiset of prime indices of n.

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A365068 Number of integer partitions of n with some part that can be written as a nonnegative linear combination of the other distinct parts.

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A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.

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A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

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A364909 Number of ways to write n as a nonnegative linear combination of a strict integer composition of n.

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A365003 Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

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A365072 Number of integer partitions of n such that no distinct part can be written as a (strictly) positive linear combination of the other distinct parts.

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