cp's OEIS Frontend

A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

%I A381870 #7 Mar 13 2025 08:55:19
%S A381870 1,2,3,5,7,11,12,13,17,18,19,20,23,28,29,31,36,37,41,43,44,45,47,50,
%T A381870 52,53,59,61,63,67,68,71,73,75,76,79,83,89,92,97,98,99,100,101,103,
%U A381870 107,109,113,116,117,120,124,127,131,137,139,147,148,149,151,153
%N A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.
%C A381870 First differs from A212166 in lacking 360.
%C A381870 First differs from A293511 in having 600.
%C A381870 Also numbers with a unique factorization into squarefree numbers with distinct sums of prime indices (A056239).
%C A381870 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.
%e A381870 For n = 600 the unique multiset partition is {{1},{1,3},{1,2,3}}. The unique factorization is 2*10*30.
%t A381870 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A381870 sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A381870 Select[Range[100],Length[Select[sfacs[#],UnsameQ@@hwt/@#&]]==1&]
%Y A381870 Without distinct block-sums we have A000961, ones in A050320.
%Y A381870 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381870 For distinct blocks instead of sums we have A293511, ones in A050326.
%Y A381870 These are the positions of ones in A381633, see A381634, A381806, A381990.
%Y A381870 Normal multiset partitions of this type are counted by A381718, see A279785.
%Y A381870 For constant instead of strict blocks we have A381991, ones in A381635.
%Y A381870 A001055 counts multiset partitions of prime indices, strict A045778.
%Y A381870 A003963 gives product of prime indices.
%Y A381870 A055396 gives least prime index, greatest A061395.
%Y A381870 A056239 adds up prime indices, row sums of A112798.
%Y A381870 A122111 represents conjugation in terms of Heinz numbers.
%Y A381870 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381870 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381870 A321469 counts factorizations with distinct sums of prime indices, ones A166684.
%Y A381870 Cf. A000720, A001222, A005117, A066328, A293243, A299202, A300385.
%K A381870 nonn
%O A381870 1,2
%A A381870 _Gus Wiseman_, Mar 12 2025