A321455 -id:A321455 - cp's OEIS frontend

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Total/@Split[#]&]]==0&]],{n,0,15}]

More terms from Bert Dobbelaere, Apr 26 2025

A336137 Number of set partitions of the binary indices of n with equal block-sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

Offset: 0

Gus Wiseman, Jul 12 2020

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(n) set partitions for n = 7, 59, 119, 367, 127:
  {123}    {12456}      {123567}      {1234679}    {1234567}
  {12}{3}  {126}{45}    {1236}{57}    {12346}{79}  {1247}{356}
           {15}{24}{6}  {156}{237}    {1249}{367}  {1256}{347}
                        {17}{26}{35}  {1267}{349}  {1346}{257}
                                      {169}{2347}  {167}{2345}
                                                   {16}{25}{34}{7}
The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:
  {{2,7},{3,6},{4,5},{9}}
  {{2,4,6},{3,9},{5,7}}
  {{2,7,9},{3,4,5,6}}
  {{2,3,4,9},{5,6,7}}
  {{2,3,6,7},{4,5,9}}
  {{2,4,5,7},{3,6,9}}
  {{2,3,4,5,6,7,9}}
so a(382) = 7.

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

These set partitions are counted by A035470.
The version for twice-partitions is A279787.
The version for partitions of partitions is A305551.
The version for factorizations is A321455.
The version for normal multiset partitions is A326518.
The version for distinct block-sums is A336138.
Set partitions of binary indices are A050315.
Normal multiset partitions with equal lengths are A317583.
Normal multiset partitions with equal averages are A326520.
Multiset partitions with equal block-sums are ranked by A326534.
Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[bpe[n]],SameQ@@Total/@#&]],{n,0,100}]

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Gus Wiseman, Apr 22 2025

nonn

Differs from A059404, A323055, A376250 in lacking 150.
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, sum A056239.
Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

			The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.

Twice-partitions of this type (sets with a common sum) are counted by A279788.
These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
For distinct block-sums we have A381806, counted by A381990 (complement A381992).
For constant blocks we have A381871 (zeros of A381995), counted by A381993.
Partitions of this type are counted by A381994.
These are the zeros of A382080.
Normal multiset partitions of this type are counted by A382429, see A326518.
The complement counted by A383308.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A381633 counts set systems with distinct sums, see A381634, A293243.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

prix[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]]]];
Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]

A382080 Number of ways to partition the prime indices of n into sets with a common sum.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 20 2025

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, sum A056239.

Also the number of factorizations of n into squarefree numbers > 1 with equal sums of prime indices.

			The prime indices of 900 are {1,1,2,2,3,3}, with the following partitions into sets with a common sum:
  {{1,2,3},{1,2,3}}
  {{3},{3},{1,2},{1,2}}
So a(900) = 2.

For just sets we have A050320, distinct A050326.
Twice-partitions of this type are counted by A279788.
For just a common sum we have A321455.
MM-numbers of these multiset partitions are A326534 /\ A302478.
For distinct instead of equal sums we have A381633.
For constant instead of strict blocks we have A381995.
Positions of 0 are A381719, counted by A381994.
A000688 counts factorizations into prime powers, distinct A050361.
A001055 counts factorizations, strict A045778.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A006171, A279784, A353866, A381635, A381871.

prix[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[prix[n]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]],{n,100}]

A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

Original entry on oeis.org

1, 0, 1, 1, 4, 13, 91, 1318, 73581, 51913025

Offset: 0

Gus Wiseman, Jul 13 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(2) = 1 through a(5) = 13 antichains:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}      {{1,2,3,4,5}}
                      {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                      {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                      {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                       {{1,4,5},{1,2,3,4}}
                                       {{2,3,5},{1,2,3,4}}
                                       {{2,4,5},{1,2,3,5}}
                                       {{3,4,5},{1,2,4,5}}
                                       {{1,5},{2,4},{1,2,3}}
                                       {{2,5},{3,4},{1,2,4}}
                                       {{3,5},{1,2,5},{1,3,4}}
                                       {{4,5},{1,3,5},{2,3,4}}
                                       {{1,4,5},{2,3,5},{1,2,3,4}}

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326360, A326518, A326534, A326566.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n],{2,n}],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 14 2019

A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 1, 1, 2, 4, 14, 92, 1320, 73584, 51913039

Offset: 0

Gus Wiseman, Jul 13 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. It is covering if its union is {1..n}. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(1) = 1 through a(5) = 14 antichains:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{3},{1,2}}  {{1,4},{2,3}}    {{1,2,5},{1,3,4}}
                               {{2,4},{1,2,3}}  {{1,3,5},{2,3,4}}
                               {{3,4},{1,2,4}}  {{1,4,5},{2,3,5}}
                                                {{5},{1,4},{2,3}}
                                                {{1,4,5},{1,2,3,4}}
                                                {{2,3,5},{1,2,3,4}}
                                                {{2,4,5},{1,2,3,5}}
                                                {{3,4,5},{1,2,4,5}}
                                                {{1,5},{2,4},{1,2,3}}
                                                {{2,5},{3,4},{1,2,4}}
                                                {{3,5},{1,2,5},{1,3,4}}
                                                {{4,5},{1,3,5},{2,3,4}}
                                                {{1,4,5},{2,3,5},{1,2,3,4}}

The non-covering case is A326574.

Cf. A000372, A006126, A035470, A307249, A321455, A321717, A321718, A326518, A326534, A326565.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleq[n_]:=Select[stableSets[Subsets[Range[n]],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&],Union@@#==Range[n]&];
Table[Length[cleq[n]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 14 2019

A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 2, 4, 2, 5, 1, 3, 2, 4, 1, 5, 1, 2, 3, 3, 2, 5, 1, 5, 2, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 10 2025

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 prime indices of 84 are {1,1,2,4}, with 7 multiset partitions into blocks with distinct sums:
  {{1,1,2,4}}
  {{1},{1,2,4}}
  {{2},{1,1,4}}
  {{1,1},{2,4}}
  {{1,2},{1,4}}
  {{1},{2},{1,4}}
  {{1},{4},{1,2}}
with block-sums: {8}, {1,7}, {2,6}, {2,6}, {3,5}, {1,2,5}, {1,3,4}, of which 6 are distinct, so a(84) = 6.

Allowing any block-sums gives A317141  (lower A300383), before sums A001055.
Before taking sums we had A321469.
For distinct blocks instead of distinct block-sums we have A381452.
If each block is a set we have A381634 (zeros A381806), before sums A381633.
For equal instead of distinct block-sums we have A381872, before sums A321455.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A045778, A066328, A213385, A299200, A299201, A299202, A300385, A317142.

prix[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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@Total/@#&]]],{n,100}]

A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

Original entry on oeis.org

2, 3, 5, 10, 22, 61, 247, 2096, 81896, 52260575

Offset: 0

Gus Wiseman, Jul 18 2019

An antichain is a finite set of finite sets, none of which is a subset of any other. The edge-sums are the sums of vertices in each edge, so for example the edge sums of {{1,3},{2,5},{3,4,5}} are {4,7,12}.

			The a(0) = 2 through a(4) = 22 antichains:
  {}    {}     {}       {}           {}
  {{}}  {{}}   {{}}     {{}}         {{}}
        {{1}}  {{1}}    {{1}}        {{1}}
               {{2}}    {{2}}        {{2}}
               {{1,2}}  {{3}}        {{3}}
                        {{1,2}}      {{4}}
                        {{1,3}}      {{1,2}}
                        {{2,3}}      {{1,3}}
                        {{1,2,3}}    {{1,4}}
                        {{3},{1,2}}  {{2,3}}
                                     {{2,4}}
                                     {{3,4}}
                                     {{1,2,3}}
                                     {{1,2,4}}
                                     {{1,3,4}}
                                     {{2,3,4}}
                                     {{1,2,3,4}}
                                     {{3},{1,2}}
                                     {{4},{1,3}}
                                     {{1,4},{2,3}}
                                     {{2,4},{1,2,3}}
                                     {{3,4},{1,2,4}}

Set partitions with equal block-sums are A035470.
Antichains with different edge-sums are A326030.
MM-numbers of multiset partitions with equal part-sums are A326534.
The covering case is A326566.
Cf. A000372, A003182, A006126, A307249, A321455, A321717, A321718, A326518, A326565, A326572.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
cleqset[set_]:=stableSets[Subsets[set],SubsetQ[#1,#2]||Total[#1]!=Total[#2]&];
Table[Length[cleqset[Range[n]]],{n,0,5}]

a(9) from Andrew Howroyd, Aug 13 2019

A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4

Offset: 1

Gus Wiseman, Apr 13 2024

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. Sum of prime indices is given by A056239.

Factorizations into factors all having the same sum of prime indices are counted by A321455.

			The factorizations of 588 of this type are (7*7*12), (21*28), (588), so a(588) = 3.
The factorizations of 900 of this type are (5*5*6*6), (9*10*10), (25*36), (30*30), (900), so a(900) = 4.

Positions of 1's are A321453, counted by A321451.
Positions of terms > 1 are A321454, counted by A321452.
Factorizations of this type are counted by A321455, different sums A321469.
For different sums instead of same sums we have A371734.
For set partitions of binary indices we have A371735.
A001055 counts factorizations.
A002219 (aerated) counts biquanimous partitions, ranks A357976.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A321142 and A371794 count non-biquanimous strict partitions.
A371789 counts non-quanimous sets, differences A371790.
A371796 counts quanimous sets, differences A371797.
Cf. A035470, A279787, A305551, A322794, A326515, A326518, A326534, A336137, A371783, A371791.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&, Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Table[Max[Length/@Select[facs[n],SameQ@@hwt/@#&]],{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_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A371733(n, m=n, facs=List([])) = if(1==n, if(all_have_same_sum_of_pis(facs),#facs,0), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s = max(s, A371733(n/d, d, newfacs)))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A382215 MM-numbers of multiset partitions into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 35, 41, 49, 53, 59, 64, 67, 81, 83, 97, 103, 109, 121, 125, 127, 128, 131, 157, 175, 179, 191, 209, 211, 227, 241, 243, 245, 256, 277, 283, 289, 311, 331, 343, 353, 361, 367, 391, 401, 419, 431, 461

Offset: 1

Gus Wiseman, Mar 21 2025

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, sum A056239. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  31: {{5}}
  32: {{},{},{},{},{}}
  35: {{2},{1,1}}
  41: {{6}}
  49: {{1,1},{1,1}}
  53: {{1,1,1,1}}
  59: {{7}}

Twice-partitions of this type are counted by A279789.
For just constant blocks we have A302492, counted by A000688.
For sets of constant multisets we have A302496, counted by A050361.
For just common sums we have A326534, counted by A321455.
Factorizations of this type are counted by A381995.
For strict blocks and distinct sums we have A382201, counted by A381633.
Normal multiset partitions of this type are counted by A382204.
For strict instead of constant blocks we have A382304, counted by A382080.
For sets of constant multisets with distinct sums A382426, counted by A381635.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A000720, A000961, A001055, A302242, A302601, A368100, A381715, A381716, A381636, A381871.

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

is(k) = my(f=factor(k)[, 1]~, k, p, v=vector(#f, i, primepi(f[i]))); for(i=1, #v, k=isprimepower(v[i], &p); if(k||v[i]==1, v[i]=k*primepi(p), return(0))); #Set(v)<2; \\ Jinyuan Wang, Apr 02 2025

Equals A326534 /\ A302492.

cp's OEIS Frontend

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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A336137 Number of set partitions of the binary indices of n with equal block-sums.

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A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

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A382080 Number of ways to partition the prime indices of n into sets with a common sum.

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A326565 Number of covering antichains of nonempty, non-singleton subsets of {1..n}, all having the same sum.

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A326566 Number of covering antichains of subsets of {1..n} with equal edge-sums.

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A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

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A326574 Number of antichains of subsets of {1..n} with equal edge-sums.

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A371733 Maximal length of a factorization of n into factors > 1 all having the same sum of prime indices.

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