cp's OEIS Frontend

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.

%I A381637 #6 Mar 11 2025 08:24:32
%S A381637 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,
%T A381637 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,
%U A381637 2,4,1,5,1,2,3,3,2,5,1,5,2,2,1,6,2,2,2
%N 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.
%C A381637 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 A381637 The prime indices of 84 are {1,1,2,4}, with 7 multiset partitions into blocks with distinct sums:
%e A381637   {{1,1,2,4}}
%e A381637   {{1},{1,2,4}}
%e A381637   {{2},{1,1,4}}
%e A381637   {{1,1},{2,4}}
%e A381637   {{1,2},{1,4}}
%e A381637   {{1},{2},{1,4}}
%e A381637   {{1},{4},{1,2}}
%e A381637 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.
%t A381637 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381637 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381637 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381637 Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@Total/@#&]]],{n,100}]
%Y A381637 Allowing any block-sums gives A317141  (lower A300383), before sums A001055.
%Y A381637 Before taking sums we had A321469.
%Y A381637 For distinct blocks instead of distinct block-sums we have A381452.
%Y A381637 If each block is a set we have A381634 (zeros A381806), before sums A381633.
%Y A381637 For equal instead of distinct block-sums we have A381872, before sums A321455.
%Y A381637 Other multiset partitions of prime indices:
%Y A381637 - For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
%Y A381637 - For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
%Y A381637 - For sets of constant multisets (A050361) see A381715.
%Y A381637 - For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
%Y A381637 A003963 gives product of prime indices.
%Y A381637 A055396 gives least prime index, greatest A061395.
%Y A381637 A056239 adds up prime indices, row sums of A112798.
%Y A381637 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381637 Cf. A000720, A001222, A001970, A002846, A045778, A066328, A213385, A299200, A299201, A299202, A300385, A317142.
%K A381637 nonn
%O A381637 1,6
%A A381637 _Gus Wiseman_, Mar 10 2025