cp's OEIS Frontend

A381716 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 constant blocks with distinct sums.

%I A381716 #5 Mar 11 2025 08:24:23
%S A381716 1,1,1,1,1,1,1,2,1,1,1,0,1,1,1,2,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,3,1,1,
%T A381716 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,0,1,1,0,4,1,1,1,1,
%U A381716 1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,0,1,1,1
%N A381716 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 constant blocks with distinct sums.
%C A381716 First differs from A381635 at a(1728) = 4, A381635(1728) = 5.
%C A381716 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 A381716 The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into constant multisets with distinct sums:
%e A381716   {{1,1,1,1,1,1},{2,2},{2}}
%e A381716   {{1,1,1,1,1},{1},{2,2,2}}
%e A381716   {{1,1,1,1,1},{1},{2,2},{2}}
%e A381716   {{1,1,1,1},{1,1},{2,2,2}}
%e A381716   {{1,1,1},{1,1},{1},{2,2,2}}
%e A381716 with block-sums: {1,5,6}, {2,4,6}, {1,2,3,6}, {1,2,4,5}, so a(1728) = 4.
%t A381716 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381716 mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
%t A381716 Table[Length[Union[Sort[Total/@#]&/@Select[Join@@@Tuples[mce/@Split[prix[n]]],UnsameQ@@Total/@#&]]],{n,100}]
%Y A381716 Without distinct sums we have A000688, after sums A381455 (upper), A381453 (lower).
%Y A381716 More on multiset partitions into constant blocks: A006171, A279784, A295935.
%Y A381716 For strict instead of constant we have A381633, before sums A381634.
%Y A381716 Before taking sums we had A381635.
%Y A381716 Positions of 0 are A381636.
%Y A381716 For distinct blocks instead of sums we have A381715.
%Y A381716 A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
%Y A381716 A055396 gives least prime index, greatest A061395.
%Y A381716 A056239 adds up prime indices, row sums of A112798.
%Y A381716 Cf. A000720, A001222, A002846, A005117, A050361, A213242, A265947, A293511, A299202, A300385, A362421.
%K A381716 nonn
%O A381716 1,8
%A A381716 _Gus Wiseman_, Mar 10 2025