cp's OEIS Frontend

A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.

%I A381719 #9 Apr 25 2025 08:45:42
%S A381719 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,90,
%T A381719 92,96,98,99,104,108,112,116,117,120,124,126,132,135,136,140,144,147,
%U A381719 148,152,153,156,160,162,164,168,171,172,175,176,184,188,189,192
%N A381719 Numbers whose prime indices cannot be partitioned into sets with a common sum.
%C A381719 Differs from A059404, A323055, A376250 in lacking 150.
%C A381719 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 A381719 Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).
%e A381719 The prime indices of 150 are {1,2,3,3}, and {{3},{3},{1,2}} is a partition into sets with a common sum, so 150 is not in the sequence.
%t A381719 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381719 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381719 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A381719 Select[Range[100],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]
%Y A381719 Twice-partitions of this type (sets with a common sum) are counted by A279788.
%Y A381719 These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.
%Y A381719 For distinct block-sums we have A381806, counted by A381990 (complement A381992).
%Y A381719 For constant blocks we have A381871 (zeros of A381995), counted by A381993.
%Y A381719 Partitions of this type are counted by A381994.
%Y A381719 These are the zeros of A382080.
%Y A381719 Normal multiset partitions of this type are counted by A382429, see A326518.
%Y A381719 The complement counted by A383308.
%Y A381719 A000041 counts integer partitions, strict A000009.
%Y A381719 A001055 counts factorizations, strict A045778.
%Y A381719 A050320 counts factorizations into squarefree numbers, see A381078, A381454.
%Y A381719 A050326 counts factorizations into distinct squarefree numbers.
%Y A381719 A055396 gives least prime index, greatest A061395.
%Y A381719 A056239 adds up prime indices, row sums of A112798.
%Y A381719 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A381719 A381633 counts set systems with distinct sums, see A381634, A293243.
%Y A381719 Set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381719 Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.
%K A381719 nonn
%O A381719 1,1
%A A381719 _Gus Wiseman_, Apr 22 2025