A365658 -id:A365658 - cp's OEIS frontend

A126796 Number of complete partitions of n.

1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 20, 31, 39, 55, 71, 100, 125, 173, 218, 291, 366, 483, 600, 784, 971, 1244, 1538, 1957, 2395, 3023, 3693, 4605, 5604, 6942, 8397, 10347, 12471, 15235, 18309, 22267, 26619, 32219, 38414, 46216, 54941, 65838, 77958, 93076, 109908

Offset: 0

Brian Hopkins, Feb 20 2007

A partition of n is complete if every number 1 to n can be represented as a sum of parts of the partition. This generalizes perfect partitions, where the representation for each number must be unique.
A partition is complete iff each part is no more than 1 more than the sum of all smaller parts. (This includes the smallest part, which thus must be 1.) - Franklin T. Adams-Watters, Mar 22 2007
For n > 0: a(n) = sum of n-th row in A261036 and also a(floor(n/2)) = A261036(n,floor((n+1)/2)). - Reinhard Zumkeller, Aug 08 2015
a(n+1) is the number of partitions of n such that each part is no more than 2 more than the sum of all smaller parts (generalizing Adams-Watters's criterion). Bijection: each partition counted by a(n+1) must contain a 1, removing that gives a desired partition of n. - Brian Hopkins, May 16 2017
A partition (x_1, ..., x_k) is complete if and only if 1, x_1, ..., x_k is a "regular sequence" (see A003513 for definition). As a result, the number of complete partitions with n parts is given by A003513(n+1). - Nathaniel Johnston, Jun 29 2023

			There are a(5) = 4 complete partitions of 5:
  [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [1, 1, 3].
G.f.: 1 = 1*(1-x) + 1*x*(1-x)*(1-x^2) + 1*x^2*(1-x)*(1-x^2)*(1-x^3) + 2*x^3*(1-x)*(1-x^2)*(1-x^3)*(1-x^4) + 2*x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)*(1-x^5) + ...
From _Gus Wiseman_, Oct 14 2023: (Start)
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (11)  (21)   (211)   (221)    (321)     (421)      (3221)
             (111)  (1111)  (311)    (2211)    (2221)     (3311)
                            (2111)   (3111)    (3211)     (4211)
                            (11111)  (21111)   (4111)     (22211)
                                     (111111)  (22111)    (32111)
                                               (31111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 301 terms from David W. Wilson)
George E. Andrews, George Beck, and Brian Hopkins, On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions, Annals of Comb., 24(2020), pp. 217-224.
George Beck and Shane Chern, Reciprocity between partitions and compositions, arXiv:2108.04363 [math.CO], 2021.
Nathaniel Johnston and Sarah Plosker, Laplacian {-1,0,1}- and {-1,1}-diagonalizable graphs, arXiv:2308.15611 [math.CO], 2023.
SeungKyung Park, Complete Partitions, Fibonacci Quarterly, Vol. 36 (1998), pp. 354-360.

Cf. A002033, A003513, A209405, A261036, A286929, A286097.
For parts instead of sums we have A000009 (sc. coverings), ranks A055932.
The strict case is A188431, complement A365831.
These partitions have ranks A325781.
First column k = 0 of A365923.
The complement is counted by A365924, ranks A365830.
Cf. A000041, A018818, A046663, A047967, A276024, A304792, A325799, A365543, A365658, A365918, A365921.

import Data.MemoCombinators (memo3, integral, Memo)
a126796 n = a126796_list !! n
a126796_list = map (pMemo 1 1) [0..] where
   pMemo = memo3 integral integral integral p
   p   0 = 1
   p s k m
     | k > min m s = 0
     | otherwise   = pMemo (s + k) k (m - k) + pMemo s (k + 1) m
-- Reinhard Zumkeller, Aug 07 2015

isCompl := proc(p,n) local m,pers,reps,f,lst,s; reps := {}; pers := combinat[permute](p); for m from 1 to nops(pers) do lst := op(m,pers); for f from 1 to nops(lst) do s := add( op(i,lst),i=1..f); reps := reps union {s}; od; od; for m from 1 to n do if not m in reps then RETURN(false); fi; od; RETURN(true); end: A126796 := proc(n) local prts, res,p; prts := combinat[partition](n); res :=0; for p from 1 to nops(prts) do if isCompl(op(p,prts),n) then res := res+1; fi; od; RETURN(res); end: for n from 1 to 40 do printf("%d %d ",n,A126796(n)); od; # R. J. Mathar, Feb 27 2007
# At the beginning of the 2nd Maple program replace the current 15 by any other positive integer n in order to obtain a(n). - Emeric Deutsch, Mar 04 2007
with(combinat): a:=proc(n) local P,b,k,p,S,j: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: end: seq(a(n),n=1..30); # Emeric Deutsch, Mar 04 2007
with(combinat): n:=15: P:=partition(n): b:=0: for k from 1 to numbpart(n) do p:=powerset(P[k]): S:={}: for j from 1 to nops(p) do S:=S union {add(p[j][i],i=1..nops(p[j]))} od: if nops(S)=n+1 then b:=b+1 else b:=b: fi: od: b; # Emeric Deutsch, Mar 04 2007

T[n_, k_] := T[n, k] = If[k <= 1, 1, If[n < 2k-1, T[n, Floor[(n+1)/2]], T[n, k-1] + T[n-k, k]]];
a[n_] := T[n, Floor[(n+1)/2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 11 2017, after Franklin T. Adams-Watters *)
nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]]; Table[Length[Select[IntegerPartitions[n], nmz[#]=={}&]],{n,0,15}] (* Gus Wiseman, Oct 14 2023 *)

{T(n,k)=if(k<=1,1,if(n<2*k-1,T(n,floor((n+1)/2)),T(n,k-1)+T(n-k,k)))}
{a(n)=T(n,floor((n+1)/2))} /* If modified to save earlier results, this would be efficient. */ /* Franklin T. Adams-Watters, Mar 22 2007 */

/* As coefficients in g.f.: */
{a(n)=local(A=[1,1]);for(i=1,n+1,A=concat(A,0);A[#A]=polcoeff(1-sum(m=1,#A,A[m]*x^m*prod(k=1,m,1-x^k +x*O(x^#A))),#A) );A[n+1]}
for(n=0,50,print1(a(n),",")) /* Paul D. Hanna, Mar 06 2012 */

G.f.: 1 = Sum_{n>=0} a(n)*x^n*Product_{k=1..n+1} (1-x^k). - Paul D. Hanna, Mar 08 2012
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 25*Pi/(24*sqrt(6))) / sqrt(n) + (25/16 - 1679*Pi^2/6912)/n). - Vaclav Kotesovec, May 24 2018, extended Nov 02 2019
a(n) = A000041(n) - A365924(n). - Gus Wiseman, Oct 14 2023

More terms from R. J. Mathar, Feb 27 2007
More terms from Emeric Deutsch, Mar 04 2007
Further terms from Franklin T. Adams-Watters and David W. Wilson, Mar 22 2007

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 5, 7, 4, 8, 2, 6, 4, 7, 2, 8, 2, 4, 6, 6, 4, 8, 2, 8, 5, 4, 2, 9, 4, 4, 4

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

			The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
Positions of first appearances are A259941.
The triangle for this rank statistic is A365658.
The semi version is A366739, sum A366738, strict A366741.
Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Comment corrected by Gus Wiseman, Aug 09 2024

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

Original entry on oeis.org

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

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).

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[#]]&]

A367225 Numbers m without a divisor whose prime indices sum to bigomega(m).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 22, 23, 25, 26, 27, 28, 29, 31, 34, 35, 37, 38, 41, 43, 44, 46, 47, 49, 52, 53, 55, 58, 59, 61, 62, 63, 65, 67, 68, 71, 73, 74, 76, 77, 79, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 99, 101, 103, 104, 106, 107, 109, 113

Offset: 1

Gus Wiseman, Nov 15 2023

nonn

Also numbers m whose prime indices do not 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 A367213.

			The prime indices of 24 are {1,1,1,2} with submultiset {1,1,2} summing to 4, so 24 is not in the sequence.
The terms together with their prime indices begin:
     3: {2}        29: {10}       58: {1,10}
     5: {3}        31: {11}       59: {17}
     7: {4}        34: {1,7}      61: {18}
    10: {1,3}      35: {3,4}      62: {1,11}
    11: {5}        37: {12}       63: {2,2,4}
    13: {6}        38: {1,8}      65: {3,6}
    14: {1,4}      41: {13}       67: {19}
    17: {7}        43: {14}       68: {1,1,7}
    19: {8}        44: {1,1,5}    71: {20}
    22: {1,5}      46: {1,9}      73: {21}
    23: {9}        47: {15}       74: {1,12}
    25: {3,3}      49: {4,4}      76: {1,1,8}
    26: {1,6}      52: {1,1,6}    77: {4,5}
    27: {2,2,2}    53: {16}       79: {22}
    28: {1,1,4}    55: {3,5}      82: {1,13}

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.
A108917 counts knapsack partitions, ranks A299702, strict A275972.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A229816 counts partitions whose length is not a part, ranks A367107.
A237667 counts sum-free partitions, ranks A364531.
A365924 counts incomplete partitions, ranks A365830.
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, A288728, A304792, A325761, A325781, A364345, A364347.

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

A365925 Number of subset-sums of strict integer partitions of n.

Original entry on oeis.org

1, 2, 2, 6, 6, 10, 17, 22, 29, 42, 59, 74, 102, 130, 171, 226, 281, 356, 454, 566, 699, 896, 1080, 1342, 1637, 2006, 2413, 2962, 3548, 4286, 5114, 6148, 7272, 8738, 10268, 12224, 14387, 16996, 19863, 23450, 27257, 31984, 37187, 43364, 50173, 58428, 67322

Offset: 0

Gus Wiseman, Sep 26 2023

nonn

This is the "not necessarily positive" version, cf. A284640.

			The a(6) = 17 ways, showing each strict partition and its subset-sums:
    (6): 0,6
   (51): 0,1,5,6
   (42): 0,2,4,6
  (321): 0,1,2,3,4,5,6

The positive case is A284640.
The non-strict version is A304792, positive case A276024.
Row sums of A365661, non-strict A365543.
The complement (non-subset-sums) is A365922, non-strict A365918.
A000041 counts integer partitions, strict A000009.
A126796 counts complete partitions, ranks A325781, strict A188431.
A365923 counts partitions by non-subset-sums, strict A365545.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
Cf. A006827, A046663, A364272, A364350, A365658, A365663, A365921.

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

cp's OEIS Frontend

A126796 Number of complete partitions of n.

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A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

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

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