cp's OEIS Frontend

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

%I A382080 #6 Mar 22 2025 08:38:53
%S A382080 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,
%T A382080 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,
%U A382080 1,2,1,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,1
%N A382080 Number of ways to partition the prime indices of n into sets with a common sum.
%C A382080 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.
%C A382080 Also the number of factorizations of n into squarefree numbers > 1 with equal sums of prime indices.
%e A382080 The prime indices of 900 are {1,1,2,2,3,3}, with the following partitions into sets with a common sum:
%e A382080   {{1,2,3},{1,2,3}}
%e A382080   {{3},{3},{1,2},{1,2}}
%e A382080 So a(900) = 2.
%t A382080 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A382080 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382080 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A382080 Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]],{n,100}]
%Y A382080 For just sets we have A050320, distinct A050326.
%Y A382080 Twice-partitions of this type are counted by A279788.
%Y A382080 For just a common sum we have A321455.
%Y A382080 MM-numbers of these multiset partitions are A326534 /\ A302478.
%Y A382080 For distinct instead of equal sums we have A381633.
%Y A382080 For constant instead of strict blocks we have A381995.
%Y A382080 Positions of 0 are A381719, counted by A381994.
%Y A382080 A000688 counts factorizations into prime powers, distinct A050361.
%Y A382080 A001055 counts factorizations, strict A045778.
%Y A382080 A055396 gives least prime index, greatest A061395.
%Y A382080 A056239 adds up prime indices, row sums of A112798.
%Y A382080 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A382080 Cf. A000720, A000961, A001222, A006171, A279784, A353866, A381635, A381871.
%K A382080 nonn
%O A382080 1,30
%A A382080 _Gus Wiseman_, Mar 20 2025