A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Author

Gus Wiseman, Mar 13 2025

Keywords

• nonn

Comments

First differs from A383100 in lacking 108.

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 written as a product of prime powers with equal sums of prime indices.

Partitions of this type are counted by A381993.

Examples

			The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
		

Crossrefs

Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).

Constant blocks with distinct sums: A381635, A381716.

For distinct instead of equal sums we have A381636, counted by A381717.

Partitions of this type are counted by A381993, complement A383093.

These are the positions of 0 in A381995.

A001055 counts multiset partitions of prime indices, strict A045778.

A050361 counts multiset partitions into distinct constant blocks.

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. A000720, A000961, A001222, A265947, A321469, A381633, A381715, A381719, A381806.

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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Select[Range[100],Select[mps[prix[#]],SameQ@@Total/@#&&And@@SameQ@@@#&]=={}&]