A365924 -id:A365924 - cp's OEIS frontend

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0, 4, 0, 0, 1, 2, 0, 5, 0, 0, 1, 1, 4, 0, 8, 0, 0, 0, 1, 2, 4, 0, 10, 0, 0, 0, 2, 1, 2, 7, 0, 16, 0, 0, 0, 0, 2, 1, 3, 8, 0, 20, 0, 0, 0, 0, 2, 2, 2, 4, 12, 0, 31, 0, 0, 0, 0, 0, 2, 2, 2, 5, 14, 0

Offset: 0

Gus Wiseman, Sep 30 2023

			The partition (6,2,1,1) has subset-sums 0, 1, 2, 3, 4, 6, 7, 8, 9, 10 so is counted under T(10,5).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  0  1  2  0
   4  0  0  1  2  0
   5  0  0  1  1  4  0
   8  0  0  0  1  2  4  0
  10  0  0  0  2  1  2  7  0
  16  0  0  0  0  2  1  3  8  0
  20  0  0  0  0  2  2  2  4 12  0
  31  0  0  0  0  0  2  2  2  5 14  0
  39  0  0  0  0  0  4  2  2  3  6 21  0
  55  0  0  0  0  0  0  4  2  4  3  9 24  0
  71  0  0  0  0  0  0  5  4  2  4  5 10 34  0
Row n = 8 counts the following partitions:
  (4211)      .  .  .  (521)   (611)  (71)   (8)     .
  (41111)              (5111)         (431)  (62)
  (3311)                                     (53)
  (3221)                                     (44)
  (32111)                                    (422)
  (311111)                                   (332)
  (22211)                                    (2222)
  (221111)
  (2111111)
  (11111111)

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

Row sums are A000041.
Diagonal k = n-1 is A002865.
Column k = 1 is A126796 (complete partitions), ranks A325781.
Central diagonal n = 2k is A126796 also.
For parts instead of sums we have A339737, rank stat A339662, min A257993.
This is the triangle for the rank statistic A365920.
Latter row sums are A365924 (incomplete partitions), ranks A365830.
Column sums are A366127.
A055932 lists numbers whose prime indices cover an initial interval.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709/A238710 count partitions by least/greatest difference.
A342050/A342051 have prime indices with odd/even least gap.
A366128 gives the least non-subset-sum of prime indices.
Cf. A001522, A079068, A098743, A264401, A286469 or A286470, A339886.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Max@@Prepend[nmz[#],0]==k&]],{n,0,10},{k,0,n}]

A366754 Number of non-knapsack integer partitions of n.

Original entry on oeis.org

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).

A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 4, 0, 2, 0, 1, 0, 5, 1, 0, 3, 1, 1, 0, 8, 0, 3, 0, 3, 0, 1, 0, 10, 2, 1, 2, 2, 3, 1, 1, 0, 16, 0, 5, 0, 3, 0, 5, 0, 1, 0, 20, 2, 2, 4, 2, 6, 0, 4, 1, 1, 0, 31, 0, 6, 0, 8, 0, 5, 0, 5, 0, 1, 0, 39, 4, 4, 4, 1, 6, 6, 3, 2, 6, 1, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
Triangle begins:
   1
   1  0
   1  1  0
   2  0  1  0
   2  1  1  1  0
   4  0  2  0  1  0
   5  1  0  3  1  1  0
   8  0  3  0  3  0  1  0
  10  2  1  2  2  3  1  1  0
  16  0  5  0  3  0  5  0  1  0
  20  2  2  4  2  6  0  4  1  1  0
  31  0  6  0  8  0  5  0  5  0  1  0
  39  4  4  4  1  6  6  3  2  6  1  1  0
  55  0 13  0  8  0 12  0  6  0  6  0  1  0
  71  5  8  7  3  5  3 16  3  6  0  6  1  1  0
Row n = 6 counts the following partitions:
  (321)     (411)  .  (51)   (33)  (6)  .
  (3111)              (42)
  (2211)              (222)
  (21111)
  (111111)

Row sums are A000041.
The rank statistic counted by this triangle is A325799.
The strict case is A365545, weighted row sums A365922.
The complement (positive subset-sum) is A365658.
Weighted row sums are A365918, for positive subset-sums A304792.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A000009, A006827, A122768, A364272, A365919, A365921.

Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]

A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 0, 5, 2, 0, 0, 5, 0, 1, 0

Offset: 0

Gus Wiseman, Sep 24 2023

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Is column k = n - 7 given by A325695?

			Triangle begins:
  1
  1  0
  0  1  0
  1  0  1  0
  0  1  0  1  0
  0  0  2  0  1  0
  1  0  0  2  0  1  0
  1  0  0  0  3  0  1  0
  0  1  1  0  0  3  0  1  0
  0  0  3  0  0  0  4  0  1  0
  1  0  0  2  2  0  0  4  0  1  0
  1  0  0  0  5  0  0  0  5  0  1  0
  2  0  0  0  0  5  2  0  0  5  0  1  0
  2  0  1  0  0  0  8  0  0  0  6  0  1  0
  1  1  3  0  0  0  0  7  3  0  0  6  0  1  0
  2  0  4  0  1  0  0  0 12  0  0  0  7  0  1  0
  1  1  2  2  3  1  0  0  0 11  3  0  0  7  0  1  0
  2  0  3  0  7  0  1  0  0  0 16  0  0  0  8  0  1  0
  3  0  0  2  6  3  3  1  0  0  0 15  4  0  0  8  0  1  0
Row n = 12: counts the following partitions:
  (6,3,2,1)  .  .  .  .  (9,2,1)  (6,5,1)  .  .  (11,1)  .  (12)  .
  (5,4,2,1)              (8,3,1)  (6,4,2)        (10,2)
                         (7,4,1)                 (9,3)
                         (7,3,2)                 (8,4)
                         (5,4,3)                 (7,5)

Row sums are A000009, non-strict A000041.
The complement (positive subset-sums) is also A365545 with rows reversed.
Weighted row sums are A365922, non-strict A365918.
The non-strict version is A365923, complement A365658, rank stat A325799.
A046663 counts partitions without a subset summing to k, strict A365663.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A284640, A304792, A364272, A365921.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]],{n,0,10},{k,0,n}]

A365922 Number of non-subset-sums of strict integer partitions of n.

Original entry on oeis.org

0, 1, 2, 4, 8, 11, 18, 25, 38, 51, 70, 93, 122, 159, 206, 263, 328, 420, 514, 645, 776, 967, 1154, 1413, 1686, 2042, 2414, 2890, 3394, 4062, 4732, 5598, 6494, 7652, 8836, 10329, 11884, 13833, 15830, 18376, 20936, 24131, 27476, 31547, 35780, 40966, 46292, 52737

Offset: 1

Gus Wiseman, Sep 23 2023

nonn

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

			The a(6) = 11 ways, showing each strict partition and its non-subset-sums:
    (6): 1,2,3,4,5
   (51): 2,3,4
   (42): 1,3,5
  (321):

The complement (positive subset-sums) is A284640, non-strict A276024.
Weighted row sums of A365545, non-strict A365923.
Row sums of A365663, non-strict A046663.
The non-strict version is A365918.
The zero-full complement (subset-sums) is A365925, non-strict A304792.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k.
A365661 counts strict partitions w/ a subset summing to k.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A325799, A364272, A365658, A365919, A365921.

Table[Total[Length[Complement[Range[n], Total/@Subsets[#]]]& /@ Select[IntegerPartitions[n], UnsameQ@@#&]],{n,30}]

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}]

A367394 Number of integer partitions of n whose length is a semi-sum of the parts.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 3, 3, 6, 7, 14, 15, 25, 30, 46, 54, 80, 97, 139, 169, 229, 282, 382, 461, 607, 746, 962, 1173, 1499, 1817, 2302, 2787, 3467, 4201, 5216, 6260, 7702, 9261, 11294, 13524, 16418, 19572, 23658, 28141, 33756, 40081, 47949, 56662, 67493, 79639

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

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.

			For the partition y = (3,3,2,1) we have 4 = 3 + 1, so y is counted under a(9).
The a(2) = 1 through a(10) = 14 partitions:
  (11)  .  (211)  (221)  (321)   (421)   (521)    (621)    (721)
                         (2211)  (2221)  (2222)   (3222)   (3322)
                         (3111)  (3211)  (3221)   (3321)   (3331)
                                         (3311)   (4221)   (4222)
                                         (32111)  (4311)   (4321)
                                         (41111)  (32211)  (5221)
                                                  (42111)  (5311)
                                                           (32221)
                                                           (33211)
                                                           (42211)
                                                           (43111)
                                                           (331111)
                                                           (421111)
                                                           (511111)

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
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365543 counts partitions with a subset-sum k, strict A365661.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A088809, A093971, A126796, A238628, A304792, A363225, A364534, A365541, A365924, A367402.

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

A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

			The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
  (6)    (7)    (8)      (9)      (10)     (11)     (12)       (13)
  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)    (7,5)      (7,6)
  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)    (7,4)    (8,4)      (8,5)
         (6,1)  (7,1)    (7,2)    (8,2)    (8,3)    (9,3)      (9,4)
                (4,3,1)  (8,1)    (9,1)    (9,2)    (10,2)     (10,3)
                         (4,3,2)  (5,3,2)  (10,1)   (11,1)     (11,2)
                         (5,3,1)  (5,4,1)  (5,4,2)  (5,4,3)    (12,1)
                                  (6,3,1)  (6,3,2)  (6,4,2)    (6,4,3)
                                           (6,4,1)  (6,5,1)    (6,5,2)
                                           (7,3,1)  (7,3,2)    (7,4,2)
                                                    (7,4,1)    (7,5,1)
                                                    (8,3,1)    (8,3,2)
                                                    (5,4,2,1)  (8,4,1)
                                                               (9,3,1)
                                                               (6,4,2,1)

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
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A365924 counts incomplete partitions, strict A365831.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367403 counts partitions without covering semi-sums, strict A367411.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A188431, A238628, A237113, A363225, A364272, A365658, A365918, A365925, A367410.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,15}]

A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

Original entry on oeis.org

1, 3, 9, 21, 22, 27, 63, 76, 81, 117, 147, 175, 186, 189, 243, 248, 273, 286, 290, 322, 345, 351, 399, 418, 441, 513, 516, 567, 688, 715, 729, 819, 1029, 1053, 1062, 1156, 1180, 1197, 1323, 1375, 1416, 1484, 1521, 1539, 1701, 1827, 1888, 1911, 2068, 2115, 2130

Offset: 1

Gus Wiseman, Sep 25 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 terms together with their prime indices begin:
     1: {}
     3: {2}
     9: {2,2}
    21: {2,4}
    22: {1,5}
    27: {2,2,2}
    63: {2,2,4}
    76: {1,1,8}
    81: {2,2,2,2}
   117: {2,2,6}
   147: {2,4,4}
   175: {3,3,4}
   186: {1,2,11}
   189: {2,2,2,4}
   243: {2,2,2,2,2}

The LHS is A304793, counted by A365658, with empty sets A299701.
The RHS is A325799, counted by A365923 (strict A365545).
A046663 counts partitions without a subset summing to k, strict A365663.
A056239 adds up prime indices, row sums of A112798.
A276024 counts positive subset-sums of partitions, strict A284640.
A325781 ranks complete partitions, counted by A126796.
A365830 ranks incomplete partitions, counted by A365924.
A365918 counts non-subset-sums of partitions, strict A365922.
Cf. A001223, A005117, A006827, A073491, A188431, A304792, A365831.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
smu[y_]:=Union[Total/@Rest[Subsets[y]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]

Positive integers k such that A304793(k) = A325799(k).

A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 26, 31, 38, 44, 58, 64, 81, 95, 116, 137, 166, 192, 233, 278, 330, 385, 459, 542, 636, 759, 879, 1038, 1211, 1418, 1656, 1942, 2242, 2618, 3029, 3535, 4060, 4735, 5429, 6299, 7231, 8346, 9556, 11031, 12593, 14482, 16525

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

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 partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

For parts instead of sums we have A034296, ranks A073491.
For all subset-sums we have A126796, ranks A325781, strict A188431.
The complement for parts instead of sums is A239955, ranks A073492.
The complement for all sub-sums is A365924, ranks A365830, strict A365831.
The complement is counted by A367403.
The strict case is A367410, complement A367411.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365661, A365918, A365921.

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]==Union[d])&]], {n,0,15}]

cp's OEIS Frontend

A365921 Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.

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A366754 Number of non-knapsack integer partitions of n.

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A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.

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A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.

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A365922 Number of non-subset-sums of strict integer partitions of n.

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

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A367394 Number of integer partitions of n whose length is a semi-sum of the parts.

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A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

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A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

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