A326519 -id:A326519 - cp's OEIS frontend

A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

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

Offset: 1

Gus Wiseman, Nov 11 2018

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. The sum of prime indices of n is A056239(n).

			The a(72) = 8 multiset partitions with distinct block-sums:
    {{1,1,1,2,2}}
   {{1},{1,1,2,2}}
   {{2},{1,1,1,2}}
   {{1,1},{1,2,2}}
   {{1,2},{1,1,2}}
   {{2,2},{1,1,1}}
  {{1},{2},{1,1,2}}
  {{1},{1,1},{2,2}}
Missing from this list are:
    {{1},{1},{1,2,2}}
    {{1},{1,2},{1,2}}
    {{2},{2},{1,1,1}}
    {{2},{1,1},{1,2}}
   {{1},{1},{1},{2,2}}
   {{1},{1},{2},{1,2}}
   {{1},{2},{2},{1,1}}
  {{1},{1},{1},{2},{2}}

Cf. A001055, A056239, A275780, A317144, A321455, A322794, A326514, A326515, A326516, A326519, A326535, A371734.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[mps[primeMS[n]],UnsameQ@@Sort[Total/@#]&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_different_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == #facs));
A321469(n, m=n, facs=List([])) = if(1==n, all_have_different_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321469(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

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

A275780 Number of set partitions of [n] into blocks with distinct element sums.

Original entry on oeis.org

1, 1, 2, 4, 12, 43, 160, 668, 3098, 15465, 83100, 477651, 2914505, 18795814, 127790544, 911448954, 6808162094, 53067398065, 430956571977, 3636314065247, 31841519540324, 288664242344692, 2706949104147162, 26205222185730884, 261681461422075548, 2691088457402830312

Offset: 0

Alois P. Heinz, Aug 08 2016

nonn

			a(3) = 4: 123, 13|2, 1|23, 1|2|3.
a(4) = 12: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A000110, A007837, A035470, A038041, A275781, A278339, A300335, A321469, A326512, A326513, A326519, A326535.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],UnsameQ@@Total/@#&]],{n,0,10}] (* Gus Wiseman, Jul 13 2019 *)

a(n) = A000110(n) - A275781(n).

a(17)-a(25) from Christian Sievers, Aug 20 2024

A326518 Number of normal multiset partitions of weight n where every part has the same sum.

Original entry on oeis.org

1, 1, 3, 7, 15, 31, 75, 169, 445, 1199, 3471

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(4) = 15 normal multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
             {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
             {{1},{1}}  {{1,2,2}}      {{1,1,2,2}}
                        {{1,2,3}}      {{1,1,2,3}}
                        {{2},{1,1}}    {{1,2,2,2}}
                        {{3},{1,2}}    {{1,2,2,3}}
                        {{1},{1},{1}}  {{1,2,3,3}}
                                       {{1,2,3,4}}
                                       {{1,1},{1,1}}
                                       {{1,2},{1,2}}
                                       {{1,3},{2,2}}
                                       {{1,4},{2,3}}
                                       {{2},{2},{1,1}}
                                       {{3},{3},{1,2}}
                                       {{1},{1},{1},{1}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A035470, A038041, A255906, A317583, A321455, A326517, A326519, A326520, A326521, A326534.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Total/@#&]],{n,0,5}]

a(10) from Robert Price, Apr 04 2025

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

Original entry on oeis.org

1, 1, 2, 6, 23, 106, 549, 3184, 20353, 141615, 1063399, 8554800, 73281988, 665141182, 6369920854, 64133095134, 676690490875, 7462023572238, 85786458777923, 1025956348473929, 12739037494941490

Offset: 0

Gus Wiseman, Mar 26 2025

nonn

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.

			The a(1) = 1 through a(3) = 6 multiset partitions:
  {{1}}  {{1,2}}    {{1,2,3}}
         {{1},{2}}  {{1},{1,2}}
                    {{1},{2,3}}
                    {{2},{1,2}}
                    {{2},{1,3}}
                    {{1},{2},{3}}
The a(4) = 23 factorizations:
  2*3*6  5*30    3*30    2*30    210
         10*15   6*15    6*10    2*105
         2*5*15  2*3*15  2*3*10  3*70
         3*5*10                  5*42
                                 7*30
                                 6*35
                                 10*21
                                 2*3*35
                                 2*5*21
                                 2*7*15
                                 3*5*14
                                 2*3*5*7

For distinct blocks instead of sums we have A116539, see A050326.
Without distinct sums we have A116540 (normal set multipartitions).
Twice-partitions of this type are counted by A279785.
Without strict blocks we have A326519.
Factorizations of this type are counted by A381633.
For constant instead of strict blocks we have A382203.
For distinct sizes instead of sums we have A382428, non-strict blocks A326517.
For equal instead of distinct block-sums we have A382429, non-strict blocks A326518.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A255903, A275780, A317532, A317583, A321469, A381635, A382204, A382214, A382216.

allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]

a(10)-a(11) from Robert Price, Mar 31 2025

a(12)-a(20) from Christian Sievers, Apr 05 2025

A326535 MM-numbers of multiset partitions where each part has a different sum.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, Jul 12 2019

nonn

First differs from A298540 in having 187.

These are numbers where each prime index has a different sum of prime indices. 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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

			The sequence of multiset partitions where each part has a different sum, preceded by their MM-numbers, begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  30: {{},{1},{2}}
  31: {{5}}

Robert Price, Table of n, a(n) for n = 1..9715
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A112798, A275780, A302242, A320324, A321469, A326519, A326533, A326534, A326536, A326537.

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

A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

			The a(3) = 8 multiset partitions:
  {{1,1,1}}
  {{1,1,2}}
  {{1,2,2}}
  {{1,2,3}}
  {{1},{1},{1}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Cf. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521, A331315.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Join@@mps/@allnorm[n],SameQ@@Length/@#&]],{n,8}]

\\ here U(n,m) gives number for m blocks of size n.
U(n,m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )}
a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018

a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018

a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020

Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018

A326517 Number of normal multiset partitions of weight n where each part has a different size.

Original entry on oeis.org

1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856

Offset: 0

Gus Wiseman, Jul 12 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers.

			The a(0) = 1 through a(3) = 12 normal multiset partitions:
  {}  {{1}}  {{1,1}}  {{1,1,1}}
             {{1,2}}  {{1,1,2}}
                      {{1,2,2}}
                      {{1,2,3}}
                      {{1},{1,1}}
                      {{1},{1,2}}
                      {{1},{2,2}}
                      {{1},{2,3}}
                      {{2},{1,1}}
                      {{2},{1,2}}
                      {{2},{1,3}}
                      {{3},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(i+k-1, k-1)^j, j=0..min(1, n/i))))
    end:
a:= n->add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..n), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 23 2023

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@mps/@allnorm[n],UnsameQ@@Length/@#&]],{n,0,6}]

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}
seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

A336130 Number of ways to split a strict composition of n into contiguous subsequences all having the same sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 15, 13, 23, 27, 73, 65, 129, 133, 241, 375, 519, 617, 1047, 1177, 1859, 2871, 3913, 4757, 7653, 8761, 13273, 16155, 28803, 30461, 50727, 55741, 87743, 100707, 152233, 168425, 308937, 315973, 500257, 571743, 871335, 958265, 1511583, 1621273, 2449259, 3095511, 4335385, 4957877, 7554717, 8407537, 12325993, 14301411, 20348691, 22896077, 33647199, 40267141, 56412983, 66090291, 93371665, 106615841, 155161833

Offset: 0

Gus Wiseman, Jul 11 2020

nonn

			The a(1) = 1 through a(7) = 13 splits:
  (1)  (2)  (3)    (4)    (5)    (6)        (7)
            (1,2)  (1,3)  (1,4)  (1,5)      (1,6)
            (2,1)  (3,1)  (2,3)  (2,4)      (2,5)
                          (3,2)  (4,2)      (3,4)
                          (4,1)  (5,1)      (4,3)
                                 (1,2,3)    (5,2)
                                 (1,3,2)    (6,1)
                                 (2,1,3)    (1,2,4)
                                 (2,3,1)    (1,4,2)
                                 (3,1,2)    (2,1,4)
                                 (3,2,1)    (2,4,1)
                                 (1,2),(3)  (4,1,2)
                                 (2,1),(3)  (4,2,1)
                                 (3),(1,2)
                                 (3),(2,1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with different instead of equal sums is A336128.
Starting with a non-strict composition gives A074854.
Starting with a partition gives A317715.
Starting with a strict partition gives A318683.
Set partitions with equal block-sums are A035470.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A317508, A318684, A326519, A336127, A336132, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],SameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(60) from Max Alekseyev, Feb 14 2024

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

cp's OEIS Frontend

A321469 Number of factorizations of n into factors > 1 with different sums of prime indices. Number of multiset partitions of the multiset of prime indices of n with distinct block-sums.

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A275780 Number of set partitions of [n] into blocks with distinct element sums.

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A326518 Number of normal multiset partitions of weight n where every part has the same sum.

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A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

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A326535 MM-numbers of multiset partitions where each part has a different sum.

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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size.

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A326517 Number of normal multiset partitions of weight n where each part has a different size.

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