cp's OEIS Frontend

A382075 Numbers whose prime indices can be partitioned into a set of sets with distinct sums.

%I A382075 #9 Mar 20 2025 22:35:20
%S A382075 1,2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,26,28,29,30,31,33,
%T A382075 34,35,36,37,38,39,41,42,43,44,45,46,47,50,51,52,53,55,57,58,59,60,61,
%U A382075 62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,82,83,84
%N A382075 Numbers whose prime indices can be partitioned into a set of sets with distinct sums.
%C A382075 First differs from A212167 in having 3600.
%C A382075 First differs from A335433 in lacking 72.
%C A382075 First differs from A339741 in having 1080.
%C A382075 First differs from A345172 in lacking 72.
%C A382075 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.
%C A382075 Also numbers that can be written as a product of squarefree numbers with distinct sums of prime indices.
%e A382075 The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets with distinct sums, so 1080 is in the sequence.
%t A382075 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382075 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A382075 Select[Range[100],Length[Select[mps[prix[#]], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]
%Y A382075 Twice-partitions of this type are counted by A279785, see also A358914.
%Y A382075 These are positions of terms > 0 in A381633, see A321469, A381078, A381634.
%Y A382075 For constant instead of strict blocks see A381635, A381636, A381716.
%Y A382075 Normal multiset partitions into sets with distinct sums are counted by A381718.
%Y A382075 The complement is A381806, counted by A381990.
%Y A382075 The case of a unique choice is A381870, counted by A382079, see A382078.
%Y A382075 Partitions of this type are counted by A381992.
%Y A382075 For distinct blocks instead of block-sums we have A382200, complement A293243.
%Y A382075 MM-numbers of multiset partitions into sets with distinct sums are A382201.
%Y A382075 Normal multisets of this type are counted by A382216, see also A382214.
%Y A382075 A001055 counts multiset partitions of prime indices, strict A045778.
%Y A382075 A050320 counts multiset partitions of prime indices into sets.
%Y A382075 A050326 counts multiset partitions of prime indices into distinct sets.
%Y A382075 A055396 gives least prime index, greatest A061395.
%Y A382075 A056239 adds up prime indices, row sums of A112798.
%Y A382075 A317141 counts coarsenings of prime indices, refinements A300383.
%Y A382075 Cf. A089259, A270995, A293511, A318360, A381441, A382077.
%Y A382075 Cf. A000720, A001222, A005117, A050345, A292432, A302494.
%K A382075 nonn
%O A382075 1,2
%A A382075 _Gus Wiseman_, Mar 19 2025