A365661 -id:A365661 - cp's OEIS frontend

A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 3, 3, 5, 7, 5, 5, 5, 5, 7, 11, 7, 8, 6, 8, 7, 11, 15, 11, 11, 11, 11, 11, 11, 15, 22, 15, 17, 15, 14, 15, 17, 15, 22, 30, 22, 23, 23, 22, 22, 23, 23, 22, 30, 42, 30, 33, 30, 33, 25, 33, 30, 33, 30, 42

Offset: 0

Gus Wiseman, Sep 16 2023

Rows are palindromic.

			Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   3   3   5
   7   5   5   5   5   7
  11   7   8   6   8   7  11
  15  11  11  11  11  11  11  15
  22  15  17  15  14  15  17  15  22
  30  22  23  23  22  22  23  23  22  30
  42  30  33  30  33  25  33  30  33  30  42
  56  42  45  44  44  43  43  44  44  45  42  56
  77  56  62  58  62  56  53  56  62  58  62  56  77
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (321)     (411)     (411)     (51)
  (42)      (321)     (321)     (3111)    (321)     (321)     (42)
  (411)     (3111)    (3111)    (2211)    (3111)    (3111)    (411)
  (33)      (2211)    (222)     (21111)   (222)     (2211)    (33)
  (321)     (21111)   (2211)    (111111)  (2211)    (21111)   (321)
  (3111)    (111111)  (21111)             (21111)   (111111)  (3111)
  (222)               (111111)            (111111)            (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Robert Price, Table of n, a(n) for n = 0..324

Columns k = 0 and k = n are A000041.
Central column n = 2k is A002219.
The complement is counted by A046663, strict A365663.
Row sums are A304792.
For subsets instead of partitions we have A365381.
The strict case is A365661.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A088809, A093971, A122768, A108917, A299701, A364911, A365541, A365658.

Table[Length[Select[IntegerPartitions[n],MemberQ[Total/@Subsets[#],k]&]],{n,0,15},{k,0,n}]

A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 3, 5, 3, 4, 4, 4, 4, 4, 4, 4, 7, 5, 7, 8, 7, 5, 7, 8, 7, 7, 8, 8, 7, 7, 8, 12, 9, 12, 9, 17, 9, 12, 9, 12, 14, 11, 12, 12, 13, 13, 12, 12, 11, 14, 21, 15, 19, 15, 21, 24, 21, 15, 19, 15, 21, 24, 19, 20, 19, 21, 22, 22, 21, 19, 20, 19, 24, 34, 23, 30, 24, 30, 25, 46, 25, 30, 24, 30, 23, 34

Offset: 2

N. J. A. Sloane

			For n = 4 there are two partitions (4, 2+2) with no subsum equal to 1, two (4, 3+1) with no subsum equal to 2 and two (4, 2+2) with no subsum equal to 3.
Triangle T(n,k) begins:
   1;
   1,  1;
   2,  2,  2;
   2,  2,  2,  2;
   4,  3,  5,  3,  4;
   4,  4,  4,  4,  4,  4;
   7,  5,  7,  8,  7,  5,  7;
   8,  7,  7,  8,  8,  7,  7,  8;
  12,  9, 12,  9, 17,  9, 12,  9, 12;
  ...
From _Gus Wiseman_, Oct 11 2023: (Start)
Row n = 8 counts the following partitions:
  (8)     (8)    (8)     (8)     (8)     (8)    (8)
  (62)    (71)   (71)    (71)    (71)    (71)   (62)
  (53)    (53)   (62)    (62)    (62)    (53)   (53)
  (44)    (44)   (611)   (611)   (611)   (44)   (44)
  (422)   (431)  (44)    (53)    (44)    (431)  (422)
  (332)          (422)   (521)   (422)          (332)
  (2222)         (2222)  (5111)  (2222)         (2222)
                         (332)
(End)

Alois P. Heinz, Rows n = 2..98, flattened
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Column k = 0 and diagonal k = n are both A002865.
Central diagonal n = 2k is A006827.
The complement with expanded domain is A365543.
The strict case is A365663, complement A365661.
Row sums are A365918, complement A304792.
For subsets instead of partitions we have A366320, complement A365381.
A000041 counts integer partitions, strict A000009.
A276024 counts distinct subset-sums of partitions.
A364272 counts sum-full strict partitions, sum-free A364349.
Cf. A000124, A002219, A093971, A108917, A122768, A299701, A365658.

g:= proc(n, i) option remember;
     `if`(n=0, 1, `if`(i>1, g(n, i-1), 0)+`if`(i>n, 0, g(n-i, i)))
    end:
b:= proc(n, i, s) option remember;
     `if`(0 in s or n in s, 0, `if`(n=0 or s={}, g(n, i),
     `if`(i<1, 0, b(n, i-1, s)+`if`(i>n, 0, b(n-i, i,
      select(y-> 0<=y and y<=n-i, map(x-> [x, x-i][], s)))))))
    end:
T:= (n, k)-> b(n, n, {min(k, n-k)}):
seq(seq(T(n, k), k=1..n-1), n=2..16);  # Alois P. Heinz, Jul 13 2012

g[n_, i_] := g[n, i] = If[n == 0, 1, If[i > 1, g[n, i-1], 0] + If[i > n, 0, g[n-i, i]]]; b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0 || s == {}, g[n, i], If[i < 1, 0, b[n, i-1, s] + If[i > n, 0, b[n-i, i, Select[Flatten[s /. x_ :> {x, x-i}], 0 <= # <= n-i &]]]]]]; t[n_, k_] := b[n, n, {Min[k, n-k]}]; Table[t[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 20 2013, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],FreeQ[Total/@Subsets[#],k]&]],{n,2,10},{k,1,n-1}] (* Gus Wiseman, Oct 11 2023 *)

Corrected and extended by Don Reble, Nov 04 2001

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

The complement (complete partitions) is A126796.

			The a(0) = 0 through a(8) = 12 partitions:
  .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
                  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
                  (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
                                (5,1)    (6,1)    (6,2)
                                (2,2,2)  (3,2,2)  (7,1)
                                (4,1,1)  (3,3,1)  (3,3,2)
                                         (5,1,1)  (4,2,2)
                                                  (4,3,1)
                                                  (5,2,1)
                                                  (6,1,1)
                                                  (2,2,2,2)
                                                  (5,1,1,1)

Joerg Arndt, Table of n, a(n) for n = 0..10000

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
The complement is A126796, ranks A325781, strict A188431.
These partitions have ranks A365830.
The strict case is A365831.
Row sums of A365923 without the first column, strict A365545.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A364350 counts combination-free strict partitions.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]

a(n) = A000041(n) - A126796(n).

A371795 Number of non-biquanimous integer partitions of n.

Original entry on oeis.org

0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922, 6842, 1172, 10143, 1745, 14883, 2108, 21637, 3104, 31185, 3737, 44583, 5232, 63261, 6419, 89134, 8988, 124754, 10390, 173525

Offset: 0

Gus Wiseman, Apr 07 2024

nonn

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

			The a(1) = 1 through a(8) = 8 partitions:
  (1)  (2)  (3)    (4)   (5)      (6)    (7)        (8)
            (21)   (31)  (32)     (42)   (43)       (53)
            (111)        (41)     (51)   (52)       (62)
                         (221)    (222)  (61)       (71)
                         (311)    (411)  (322)      (332)
                         (2111)          (331)      (521)
                         (11111)         (421)      (611)
                                         (511)      (5111)
                                         (2221)
                                         (3211)
                                         (4111)
                                         (22111)
                                         (31111)
                                         (211111)
                                         (1111111)

The complement is counted by A002219 aerated, ranks A357976.
Even bisection is A006827, odd A058695.
The strict complement is A237258, ranks A357854.
This is the "bi-" version of A321451, ranks A321453.
The complement is the "bi-" version of A321452, ranks A321454.
These partitions have ranks A371731.
The strict case is A371794, bisections A321142, A078408.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
A371736 counts non-quanimous strict partitons, complement A371737.
A371781 lists numbers with biquanimous prime signature, complement A371782.
A371783 counts k-quanimous partitions.
A371789 counts non-quanimous sets, differences A371790.
A371791 counts biquanimous sets, differences A232466.
A371792 counts non-biquanimous sets, differences A371793.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A064914, A305551, A336137, A365543, A365661, A365663, A366320, A365925, A367094, A371788.

biqQ[y_]:=MemberQ[Total/@Subsets[y],Total[y]/2];
Table[Length[Select[IntegerPartitions[n],Not@*biqQ]],{n,0,15}]

a(n) = if(n%2, numbpart(n), my(v=partitions(n/2), w=List([])); for(i=1, #v, for(j=1, i, listput(w, vecsort(concat(v[i], v[j]))))); numbpart(n)-#Set(w)); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 7, 4, 4, 8, 8, 14, 14, 14, 8, 8, 16, 16, 28, 28, 37, 28, 28, 16, 16, 32, 32, 56, 56, 74, 74, 74, 56, 56, 32, 32, 64, 64, 112, 112, 148, 148, 175, 148, 148, 112, 112, 64, 64, 128, 128, 224, 224, 296, 296, 350, 350, 350, 296, 296, 224, 224, 128, 128

Offset: 2

Gus Wiseman, Sep 15 2023

Rows are palindromic.

			Triangle begins:
    1
    2    2    2
    4    4    7    4    4
    8    8   14   14   14    8    8
   16   16   28   28   37   28   28   16   16
   32   32   56   56   74   74   74   56   56   32   32
Row n = 4 counts the following subsets:
  {1,2}      {1,3}      {1,4}      {2,4}      {3,4}
  {1,2,3}    {1,2,3}    {2,3}      {1,2,4}    {1,3,4}
  {1,2,4}    {1,3,4}    {1,2,3}    {2,3,4}    {2,3,4}
  {1,2,3,4}  {1,2,3,4}  {1,2,4}    {1,2,3,4}  {1,2,3,4}
                        {1,3,4}
                        {2,3,4}
                        {1,2,3,4}

Row lengths are A005408.
The case counting only length-2 subsets is A008967.
Column k = n + 1 appears to be A167762.
The version for all subsets (instead of just pairs) is A365381.
Column k = n is A365544.
A000009 counts subsets summing to n.
A007865/A085489/A151897 count certain types of sum-free subsets.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A093971/A088809/A364534 count certain types of sum-full subsets.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A068911, A095944, A238628, A288728, A326083, A364272, A365376, A365377.

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#,{2}],k]&]], {n,2,11}, {k,3,2n-1}]

A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

Offset: 0

Gus Wiseman, Sep 28 2023

nonn

			The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
The a(2) = 1 through a(10) = 9 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
            (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
                   (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
                                 (6,1)  (7,1)    (7,2)    (8,2)
                                        (4,3,1)  (8,1)    (9,1)
                                        (5,2,1)  (4,3,2)  (5,3,2)
                                                 (5,3,1)  (5,4,1)
                                                 (6,2,1)  (6,3,1)
                                                          (7,2,1)

For parts instead of sums we have ranks A080259, A055932.
The strict complement is A188431, non-strict A126796 (ranks A325781).
Row sums of A365545 without the first column, non-strict A365923.
The non-strict version is A365924, ranks A365830.
A000041 counts integer partitions, strict A000009.
A046663 counts partitions w/o a submultiset summing to k, strict A365663.
A276024 counts positive subset-sums of partitions, strict A284640.
A325799 counts non-subset-sums of prime indices.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[nmz[#]]>0&]],{n,0,15}]

A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 4, 7, 8, 12, 13, 19, 21, 29, 33, 45, 49, 67, 73, 97, 108, 139, 152, 196, 217, 274, 303, 379, 420, 523, 579, 709, 786, 960, 1061, 1285, 1423, 1714, 1885, 2265, 2498, 2966, 3280, 3881, 4268, 5049, 5548, 6507, 7170, 8391, 9194, 10744, 11778, 13677

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions are necessarily incomplete (A365924).

Are there any decreases after the initial terms?

			The a(3) = 1 through a(9) = 8 partitions:
  (3)  (4)    (5)    (6)      (7)      (8)        (9)
       (3,1)  (4,1)  (3,3)    (4,3)    (4,4)      (5,4)
                     (5,1)    (6,1)    (5,3)      (6,3)
                     (2,2,2)  (5,1,1)  (7,1)      (8,1)
                     (4,1,1)           (4,2,2)    (4,4,1)
                                       (6,1,1)    (5,2,2)
                                       (5,1,1,1)  (7,1,1)
                                                  (6,1,1,1)

Chai Wah Wu, Table of n, a(n) for n = 0..65

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213*   A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237667 counts sum-free partitions, sum-full A237668.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000700, A124506, A238628, A240861, A364349, A364531, A365045, A365381, A365918, A366320.

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

a(41)-a(54) from Chai Wah Wu, Nov 13 2023

A367224 Numbers m with a divisor whose prime indices sum to bigomega(m).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 30, 32, 33, 36, 39, 40, 42, 45, 48, 50, 51, 54, 56, 57, 60, 64, 66, 69, 70, 72, 75, 78, 80, 81, 84, 87, 90, 93, 96, 100, 102, 105, 108, 110, 111, 112, 114, 120, 123, 125, 126, 128, 129, 130, 132, 135, 138, 140, 141

Offset: 1

Gus Wiseman, Nov 14 2023

nonn

Also numbers m whose prime indices have a submultiset summing to bigomega(m).
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 A367212.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   32: {1,1,1,1,1}

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224*   A367225    A367226    A367227
A000700 counts self-conjugate partitions, ranks A088902.
A002865 counts partitions whose length is a part, ranks A325761.
A005117 ranks strict integer partitions, counted by A000009.
A066208 ranks partitions into odd parts, also counted by A000009.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A126796 counts complete partitions, ranks A325781.
A229816 counts partitions whose length is not a part, ranks A367107.
A237668 counts sum-full partitions, ranks A364532.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000720, A055396, A061395, A106529, A299729, A304792, A363225, A365830.

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

cp's OEIS Frontend

A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

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A046663 Triangle: T(n,k) = number of partitions of n (>=2) with no subsum equal to k (1 <= k <= n-1).

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A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

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A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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A371795 Number of non-biquanimous integer partitions of n.

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A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

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A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

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