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

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192

Offset: 1

Author

Gus Wiseman, Apr 22 2025

Keywords

• nonn

Comments

Differs from A059404, A323055, A376250 in lacking 150.

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.

Also numbers that cannot be factored into squarefree numbers with a common sum of prime indices (A056239).

Examples

			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.
		

Crossrefs

Twice-partitions of this type (sets with a common sum) are counted by A279788.

These multiset partitions (sets with a common sum) are ranked by A326534 /\ A302478.

For distinct block-sums we have A381806, counted by A381990 (complement A381992).

For constant blocks we have A381871 (zeros of A381995), counted by A381993.

Partitions of this type are counted by A381994.

These are the zeros of A382080.

Normal multiset partitions of this type are counted by A382429, see A326518.

The complement counted by A383308.

A000041 counts integer partitions, strict A000009.

A001055 counts factorizations, strict A045778.

A050320 counts factorizations into squarefree numbers, see A381078, A381454.

A050326 counts factorizations into distinct squarefree numbers.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A317141 counts coarsenings of prime indices, refinements A300383.

A381633 counts set systems with distinct sums, see A381634, A293243.

Set multipartitions: A089259, A116540, A270995, A296119, A318360.

Cf. A000720, A000961, A001222, A293511, A321455, A358914, A381635, A381717.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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],Select[mps[prix[#]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]=={}&]