cp's OEIS Frontend

A381715 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 distinct constant blocks.

%I A381715 #10 Mar 11 2025 08:24:18
%S A381715 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,3,1,1,
%T A381715 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,4,1,1,1,1,
%U A381715 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1
%N A381715 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 distinct constant blocks.
%C A381715 First differs from A050361 at a(1728) = 7, A050361(1728) = 8.
%C A381715 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.
%e A381715 The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into distinct constant blocks:
%e A381715   {{2,2,2},{1,1,1,1,1,1}}
%e A381715   {{1},{2,2,2},{1,1,1,1,1}}
%e A381715   {{2},{2,2},{1,1,1,1,1,1}}
%e A381715   {{1,1},{2,2,2},{1,1,1,1}}
%e A381715   {{1},{2},{2,2},{1,1,1,1,1}}
%e A381715   {{1},{1,1},{1,1,1},{2,2,2}}
%e A381715   {{2},{1,1},{2,2},{1,1,1,1}}
%e A381715   {{1},{2},{1,1},{2,2},{1,1,1}}
%e A381715 with sums:
%e A381715   {6,6}
%e A381715   {1,5,6}
%e A381715   {2,4,6}
%e A381715   {2,4,6}
%e A381715   {1,2,4,5}
%e A381715   {1,2,3,6}
%e A381715   {2,2,4,4}
%e A381715   {1,2,2,3,4}
%e A381715 of which 7 are distinct, so a(1728) = 7.
%t A381715 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381715 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381715 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381715 Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&&And@@SameQ@@@#&]]],{n,100}]
%Y A381715 Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.
%Y A381715 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381715 Positions of terms > 1 are A046099.
%Y A381715 Before taking sums we had A050361.
%Y A381715 For equal instead of distinct blocks we have A362421.
%Y A381715 For strict instead of constant blocks we have A381441, before sums A050326.
%Y A381715 For just distinct blocks we have A381452, before sums A045778.
%Y A381715 For distinct sums we have A381716, before sums A381635, zeros A381636.
%Y A381715 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
%Y A381715 A003963 gives product of prime indices.
%Y A381715 A055396 gives least prime index, greatest A061395.
%Y A381715 A056239 adds up prime indices, row sums of A112798.
%Y A381715 Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.
%K A381715 nonn
%O A381715 1,8
%A A381715 _Gus Wiseman_, Mar 10 2025