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

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, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Author

Gus Wiseman, Mar 19 2025

Keywords

• nonn

Comments

First differs from A212167 in having 3600.

First differs from A335433 in lacking 72.

First differs from A339741 in having 1080.

First differs from A345172 in lacking 72.

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.

Also numbers that can be written as a product of squarefree numbers with distinct sums of prime indices.

Examples

			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.
		

Crossrefs

Twice-partitions of this type are counted by A279785, see also A358914.

These are positions of terms > 0 in A381633, see A321469, A381078, A381634.

For constant instead of strict blocks see A381635, A381636, A381716.

Normal multiset partitions into sets with distinct sums are counted by A381718.

The complement is A381806, counted by A381990.

The case of a unique choice is A381870, counted by A382079, see A382078.

Partitions of this type are counted by A381992.

For distinct blocks instead of block-sums we have A382200, complement A293243.

MM-numbers of multiset partitions into sets with distinct sums are A382201.

Normal multisets of this type are counted by A382216, see also A382214.

A001055 counts multiset partitions of prime indices, strict A045778.

A050320 counts multiset partitions of prime indices into sets.

A050326 counts multiset partitions of prime indices into distinct sets.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A317141 counts coarsenings of prime indices, refinements A300383.

Cf. A089259, A270995, A293511, A318360, A381441, A382077.

Cf. A000720, A001222, A005117, A050345, A292432, A302494.

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Select[Range[100],Length[Select[mps[prix[#]], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]