A366740 -id:A366740 - cp's OEIS frontend

A366754 Number of non-knapsack integer partitions of n.

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 52, 60, 94, 118, 175, 213, 310, 373, 528, 643, 862, 1044, 1403, 1699, 2199, 2676, 3426, 4131, 5256, 6295, 7884, 9479, 11722, 14047, 17296, 20623, 25142, 29942, 36299, 43081, 51950, 61439, 73668, 87040, 103748, 122149, 145155, 170487

Offset: 0

Gus Wiseman, Nov 08 2023

nonn

A multiset is non-knapsack if there exist two different submultisets with the same sum.

			The a(4) = 1 through a(9) = 13 partitions:
  (211)  (2111)  (321)    (3211)    (422)      (3321)
                 (2211)   (22111)   (431)      (4221)
                 (3111)   (31111)   (3221)     (4311)
                 (21111)  (211111)  (4211)     (5211)
                                    (22211)    (32211)
                                    (32111)    (33111)
                                    (41111)    (42111)
                                    (221111)   (222111)
                                    (311111)   (321111)
                                    (2111111)  (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)

The complement is counted by A108917, strict A275972, ranks A299702.
These partitions have ranks A299729.
The strict case is A316402.
The binary version is A366753, ranks A366740.
A000041 counts integer partitions, strict A000009.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with subset-sum k, complement A046663.
A365661 counts strict partitions with subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
Cf. A002033, A006827, A122768, A126796, A238628, A365923, A365924, A367095.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#]]&]], {n,0,15}]

a(n) = A000041(n) - A108917(n).

A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 0, 3, 0, 1, 1, 1, 1, 3, 0, 1, 1, 2, 0, 3, 0, 2, 2, 1, 0, 2, 1, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 4, 0, 1, 2, 1, 1, 3, 0, 2, 1, 3, 0, 3, 0, 1, 2, 2, 1, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 1, 2, 1, 1, 1, 2, 0, 2, 2, 3, 0, 3, 0, 2, 3

Offset: 1

Gus Wiseman, Nov 04 2023

nonn

First differs from A086971 at a(90) = 3, A086971(90) = 4.
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.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The prime indices of 90 are {1,2,2,3}, with semi-sums
  3 = 1+2
  4 = 1+3 (or 2+2)
  5 = 2+3
so a(90) = 3.
Alternatively, the semiprime divisors of 90 are (6,9,10,15), with prime indices ({1,2},{2,2},{1,3},{2,3}) with sums (3,4,4,5) so a(90) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

The non-binary version is A299701.
Summing over partitions gives A366738, strict A366741.
For all sums of pairs of elements we have A367095.
Positions of first appearances are A367097.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299702 ranks knapsack partitions, counted by A108917.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000005, A000720, A001248, A008967, A117855, A304792, A304793, A365541, A365920, A366740, A366753, A367093.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Total/@Subsets[prix[n],{2}]]],{n,100}]

A366739(n) = #Set(apply(d->((f)->sum(i=1,#f~,f[i,2]*primepi(f[i,1])))(factor(d)), select(d->2==bigomega(d), divisors(n)))); \\ Antti Karttunen, Jan 20 2025

a(n) <= A086971(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 5, 1, 3, 3, 1, 3, 6, 1, 3, 3, 6, 1, 3, 1, 3, 3, 3, 3, 6, 1, 3, 1, 3, 1, 6, 3, 3, 3, 3, 1, 5, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 5

Offset: 1

Gus Wiseman, Nov 06 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.

Is the image missing only 2 and 4?

			The prime indices of 15 are {2,3}, with sums of pairs:
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(15) = 3.
The prime indices of 180 are {1,1,2,2,3}, with sums of pairs:
  1+1 = 2
  1+2 = 3
  1+3 = 4
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(180) = 5.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to prime indices in the factorization of n.

Depends only on squarefree kernel A007947. (Even more exactly, on A322591 - Antti Karttunen, Jan 20 2025)
Positions of first appearances appear to be a subset of A325986.
For 2-element submultisets we have A366739, for all submultisets A299701.
A001222 counts prime factors (also indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A304793 counts positive subset-sums of prime indices.
A367096 lists semiprime divisors, count A086971.
Cf. A001248, A004526, A008967, A366737, A366738, A366740, A367093, A367097.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Total/@Tuples[prix[n],2]]],{n,100}]

A367095(n) = if(1==n, 0, my(pis=apply(primepi,factor(n)[,1]), pairsums = vector(binomial(1+#pis,2)), k=0); for(i=1,#pis,for(j=i,#pis,k++; pairsums[k] = pis[i]+pis[j])); #Set(pairsums)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(105) by Antti Karttunen, Jan 20 2025

A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

Original entry on oeis.org

1, 90, 630, 2310, 6930, 34650, 30030, 90090, 450450, 570570, 510510, 1531530, 7657650, 14804790, 11741730, 9699690, 29099070, 145495350

Offset: 0

Gus Wiseman, Nov 05 2023

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.
We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
Are all primorials after 210 included?

			The terms together with their prime indices begin:
       1: {}
      90: {1,2,2,3}
     630: {1,2,2,3,4}
    2310: {1,2,3,4,5}
    6930: {1,2,2,3,4,5}
   34650: {1,2,2,3,3,4,5}
   30030: {1,2,3,4,5,6}
   90090: {1,2,2,3,4,5,6}
  450450: {1,2,2,3,3,4,5,6}
  570570: {1,2,3,4,5,6,8}
  510510: {1,2,3,4,5,6,7}

The first part (semiprime divisors) is A086971, firsts A220264.
The second part (semi-sums of prime indices) is A366739, firsts A367097.
All sums of pairs of prime indices are counted by A367095.
The non-binary version is A367105.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A056239 adds up prime indices, row sums of A112798.
A299701 counts subset-sums of prime indices, positive A304793.
Semiprime divisors are listed by A367096 and have:
- square count: A056170
- sum: A076290
- squarefree count: A079275
- count: A086971
- firsts: A220264
Cf. A000720, A001248, A008967, A117855, A304792, A365541, A365920, A366738, A366740, A366753.

nn=10000;
w=Table[Length[Union[Subsets[prix[n],{2}]]]-Length[Union[Total/@Subsets[prix[n],{2}]]],{n,nn}];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
Table[Position[w,k][[1,1]],{k,0,spnm[w]}]

from itertools import count
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A367093(n):
    for k in count(1):
        c, a = 0, set()
        for s in (sum(p) for p in multiset_combinations({primepi(i):j for i,j in factorint(k).items()},2)):
            if s not in a:
                a.add(s)
            else:
                c += 1
            if c > n:
                break
        if c == n:
            return k # Chai Wah Wu, Nov 13 2023

a(n) is the least positive integer such that A086971(a(n)) - A366739(a(n)) = n.

a(12)-a(16) from Chai Wah Wu, Nov 13 2023

a(17) from Chai Wah Wu, Nov 18 2023

A366753 Number of integer partitions of n without all different sums of two-element submultisets.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 9, 11, 22, 27, 48, 61, 98, 123, 188, 237, 345, 435, 611, 765, 1046, 1305, 1741, 2165, 2840, 3502, 4527, 5562, 7083, 8650, 10908, 13255, 16545, 20016, 24763, 29834, 36587, 43911, 53514, 63964, 77445, 92239, 111015, 131753

Offset: 0

Gus Wiseman, Nov 07 2023

nonn

			The two-element submultisets of y = {1,1,1,2,2,3} are {1,1}, {1,2}, {1,3}, {2,2}, {2,3}, with sums 2, 3, 4, 4, 5, which are not all different, so y is counted under a(10).
The a(8) = 1 through a(13) = 11 partitions:
  (3221)  (32211)  (4321)    (33221)    (4332)      (43321)
                   (32221)   (43211)    (5331)      (53221)
                   (322111)  (322211)   (5421)      (53311)
                             (3221111)  (43221)     (54211)
                                        (322221)    (332221)
                                        (332211)    (432211)
                                        (432111)    (3222211)
                                        (3222111)   (3322111)
                                        (32211111)  (4321111)
                                                    (32221111)
                                                    (322111111)

Semiprime divisors are counted by A086971, distinct sums A366739.
The non-binary complement is A108917, strict A275972, ranks A299702.
These partitions have ranks A366740.
The non-binary version is A366754, strict A316402, ranks A299729.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sum of partitions, strict A365925.
A365543 counts partitions with a subset-sum k, complement A046663.
A365661 counts strict partitions with a subset-sum k, complement A365663.
A366738 counts semi-sums of partitions, strict A366741.
A367096 lists semiprime divisors, row sums A076290.
Cf. A002033, A001358, A006827, A008967, A122768, A126796, A365923, A365924, A367093, A367095.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Total/@Union[Subsets[#,{2}]]&]],{n,0,30}]

A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

Original entry on oeis.org

4, 6, 4, 9, 10, 4, 6, 14, 15, 4, 6, 9, 4, 10, 21, 22, 4, 6, 25, 26, 9, 4, 14, 6, 10, 15, 4, 33, 34, 35, 4, 6, 9, 38, 39, 4, 10, 6, 14, 21, 4, 22, 9, 15, 46, 4, 6, 49, 10, 25, 51, 4, 26, 6, 9, 55, 4, 14, 57, 58, 4, 6, 10, 15, 62, 9, 21, 4, 65, 6, 22, 33, 4, 34

Offset: 1

Gus Wiseman, Nov 08 2023

On the first interpretation, the first three rows are empty. On the second, the first row is (4).

			The semiprime divisors of 30 are {6,10,15}, so row 30 is (6,10,15). Without empty rows, this is row 19.
Triangle begins (empty rows indicated by dots):
   1: .
   2: .
   3: .
   4: 4
   5: .
   6: 6
   7: .
   8: 4
   9: 9
  10: 10
  11: .
  12: 4,6
Without empty rows:
   1: 4
   2: 6
   3: 4
   4: 9
   5: 10
   6: 4,6
   7: 14
   8: 15
   9: 4
  10: 6,9
  11: 4,10
  12: 21

For all divisors we have A027750.
Square terms are counted by A056170.
Row sums are A076290.
Squarefree terms are counted by A079275.
Row lengths are A086971, firsts A220264.
A000005 counts divisors.
A001222 counts prime factors (or prime indices), distinct A001221.
A001358 lists semiprimes, squarefree A006881, complement A100959.
Cf. A008967, A365829, A366738, A366739, A366740, A367093, A367098.

Table[Select[Divisors[n],PrimeOmega[#]==2&],{n,100}]

row(n) = select(x -> bigomega(x) == 2, divisors(n)); \\ Amiram Eldar, May 02 2025

A367396 Number of subsets of {1..n} whose cardinality is the sum of two distinct elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 7, 17, 40, 90, 199, 435, 939, 2007, 4258, 8976, 18817, 39263, 81595, 168969, 348820, 718134, 1474863, 3022407, 6181687, 12621135, 25727686, 52369508, 106460521, 216162987, 438431215, 888359841, 1798371648, 3637518354, 7351824439, 14848255803

Offset: 0

Gus Wiseman, Nov 21 2023

nonn

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is not counted under a(8).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}      {1,2,3}
                    {1,2,4}    {1,2,4}      {1,2,4}
                    {1,2,3,4}  {1,2,5}      {1,2,5}
                               {1,2,3,4}    {1,2,6}
                               {1,2,3,5}    {1,2,3,4}
                               {1,3,4,5}    {1,2,3,5}
                               {1,2,3,4,5}  {1,2,3,6}
                                            {1,3,4,5}
                                            {1,3,4,6}
                                            {1,3,5,6}
                                            {1,2,3,4,5}
                                            {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}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396*  A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A236912, A237113, A237667, A237668, A304792, A363225, A364272, A365658, A366131, A366740.

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

from itertools import combinations
def A367396(n): return sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 4*a(n-1) - 5*a(n-2) + 4*a(n-3) - 5*a(n-4) + 2*a(n-5) for n > 4.
G.f.: x^3*(x - 1)/((2*x - 1)*(x^4 - 2*x^3 + x^2 - 2*x + 1)). (End)

a(18)-a(33) from Chai Wah Wu, Nov 21 2023

a(34) from Paul Muljadi, Nov 24 2023

A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

Original entry on oeis.org

4, 12, 18, 30, 36, 40, 42, 54, 60, 66, 78, 81, 90, 100, 102, 112, 114, 120, 126, 135, 138, 140, 150, 168, 174, 180, 186, 189, 198, 210, 220, 222, 225, 234, 246, 250, 252, 258, 260, 270, 280, 282, 297, 300, 306, 315, 318, 330, 336, 340, 342, 350, 351, 352, 354

Offset: 1

Gus Wiseman, Nov 21 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.

These are the Heinz numbers of the partitions counted by A367394.

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397*  A367401
A325761 ranks partitions whose length is a part, counted by A002865.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A304792 counts subset-sums of partitions, strict A365925.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A229816, A237667, A237668, A238628, A363225, A364272, A365543, A365658, A365918, A366740.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MemberQ[Total/@Subsets[prix[#],{2}],PrimeOmega[#]]&]

A367400 Number of subsets of {1..n} whose cardinality is not the sum of two distinct elements.

Original entry on oeis.org

1, 2, 4, 7, 13, 25, 47, 88, 166, 313, 589, 1109, 2089, 3934, 7408, 13951, 26273, 49477, 93175, 175468, 330442, 622289, 1171897, 2206921, 4156081, 7826746, 14739356, 27757207, 52272469, 98439697, 185381983, 349112000, 657448942, 1238110153

Offset: 0

Gus Wiseman, Nov 21 2023

			The set s = {1,2,3,6,7,8} has the following sums of pairs of distinct elements: {3,4,5,7,8,9,10,11,13,14,15}. This does not include 6, so s is counted under a(8).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,3,4}
                         {2,3,4}

The version containing n appears to be A112575.
The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400*
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A002865 counts partitions whose length is a part, complement A229816.
A364534 counts sum-full subsets.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A237113, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366131, A366740.

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

from itertools import combinations
def A367400(n): return (n*(n+1)>>1)+1+sum(1 for k in range(3,n+1) for w in (set(d) for d in combinations(range(1,n+1),k)) if not any({a,k-a}<=w for a in range(1,k+1>>1))) # Chai Wah Wu, Nov 21 2023

Conjectures from Chai Wah Wu, Nov 21 2023: (Start)
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n > 3.
G.f.: (-x^3 + x^2 + 1)/(x^4 - 2*x^3 + x^2 - 2*x + 1). (End)

a(18)-a(33) from Chai Wah Wu, Nov 21 2023

A367401 Numbers m such that bigomega(m) is not the sum of prime indices of any semiprime divisor of m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Gus Wiseman, Nov 21 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.

These are the Heinz numbers of the partitions counted by A367398.

			60 has semiprime divisor 10 with prime indices {1,3} summing to 4 = bigomega(60), so 60 is not in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401*
A002865 counts partitions w/ length, complement A229816, ranks A325761.
A088809 and A093971 count subsets containing semi-sums.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237113 counts partitions with a semi-sum of the parts, ranks A364462.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A365381 counts subsets with a subset summing to k, complement A366320.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A237667, A238628, A304792, A363225, A364272, A365543, A365658, A365918, A366740.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], FreeQ[Total/@Subsets[prix[#],{2}], PrimeOmega[#]]&]

cp's OEIS Frontend

A366754 Number of non-knapsack integer partitions of n.

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A366739 Number of distinct semi-sums of the multiset of prime indices of n. Number of distinct sums of prime indices of semiprime divisors of n (counted by A086971).

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A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

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A367093 Least positive integer with n more semiprime divisors than semi-sums of prime indices.

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A366753 Number of integer partitions of n without all different sums of two-element submultisets.

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A367096 Irregular triangle read by rows where row n lists the semiprime divisors of n. Alternatively, row n lists the semiprime divisors of A002808(n).

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A367396 Number of subsets of {1..n} whose cardinality is the sum of two distinct elements.

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A367397 Numbers m such that bigomega(m) is the sum of prime indices of some semiprime divisor of m.

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