cp's OEIS Frontend

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

%I A381995 #16 Apr 25 2025 23:40:40
%S A381995 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,
%T A381995 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,
%U A381995 0,0,1,0,1,0,0,0,0,0,1,0,3,0,1,0,0,0,0
%N A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.
%C A381995 Also the number of factorizations of n into prime powers > 1 with equal sums of prime indices.
%F A381995 A323774(n) = Sum_{A056239(k)=n} a(k). _Gus Wiseman_, Apr 25 2025
%e A381995 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:
%e A381995   {{2,2},{1,1,1,1}}
%e A381995   {{2},{2},{1,1},{1,1}}
%e A381995 so a(144) = 2.
%t A381995 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381995 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381995 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A381995 Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@SameQ@@@#&]],{n,100}]
%Y A381995 For just constant blocks we have A000688.
%Y A381995 Twice-partitions of this type are counted by A279789.
%Y A381995 For just a common sum we have A321455.
%Y A381995 For distinct instead of equal sums we have A381635.
%Y A381995 Positions of 0 are A381871, counted by A381993.
%Y A381995 MM-numbers of these multiset partitions are A382215.
%Y A381995 A001055 counts factorizations, strict A045778.
%Y A381995 A050361 counts factorizations into distinct prime powers.
%Y A381995 A055396 gives least prime index, greatest A061395.
%Y A381995 A056239 adds up prime indices, row sums of A112798.
%Y A381995 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381995 A353864 counts rucksack partitions, ranked by A353866.
%Y A381995 Cf. A279784, A295935, A381453 (lower), A381455 (upper).
%Y A381995 Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.
%Y A381995 Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.
%K A381995 nonn
%O A381995 1,4
%A A381995 _Gus Wiseman_, Mar 19 2025