A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

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

Offset: 1

Author

Gus Wiseman, Mar 19 2025

Keywords

• nonn

Comments

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

Examples

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 2 multiset partitions into constant blocks with a common sum:
  {{2,2},{1,1,1,1}}
  {{2},{2},{1,1},{1,1}}
so a(144) = 2.
		

Crossrefs

For just constant blocks we have A000688.

Twice-partitions of this type are counted by A279789.

For just a common sum we have A321455.

For distinct instead of equal sums we have A381635.

Positions of 0 are A381871, counted by A381993.

MM-numbers of these multiset partitions are A382215.

A001055 counts factorizations, strict A045778.

A050361 counts factorizations into distinct prime powers.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A317141 counts coarsenings of prime indices, refinements A300383.

A353864 counts rucksack partitions, ranked by A353866.

Cf. A279784, A295935, A381453 (lower), A381455 (upper).

Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.

Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.

Programs

Mathematica

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@@SameQ@@@#&]],{n,100}]

Formula

A323774(n) = Sum_{A056239(k)=n} a(k). Gus Wiseman, Apr 25 2025