A381806 Numbers that cannot be written as a product of squarefree numbers with distinct sums of prime indices.

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Author

Gus Wiseman, Mar 12 2025

Keywords

• nonn

Comments

First differs from A212164 in having 3600.

First differs from A293243 in having 18000.

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 whose prime indices cannot be partitioned into a multiset of sets with distinct sums.

Examples

			There are 4 factorizations of 18000 into squarefree numbers:
  (2*2*3*5*10*30)
  (2*2*5*6*10*15)
  (2*2*10*15*30)
  (2*5*6*10*30)
but none of these has all distinct sums of prime indices, so 18000 is in the sequence.
		

Crossrefs

Strongly normal multisets of this type are counted by A292444.

These are the zeros in A381633, see A050320, A321469, A381078, A381634.

For distinct blocks see A050326, A293243, A293511, A358914, A381441.

For more on set multipartitions see A089259, A116540, A270995, A296119, A318360.

For more on set multipartitions with distinct sums see A279785, A381718.

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

Partitions of this type are counted by A381990, complement A381992.

The complement is A382075.

A001055 counts multiset partitions, strict A045778.

A003963 gives product of prime indices.

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. A000688, A000720, A001222, A005117, A299202, A300385, A381454, A381870.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfics[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfics[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]]
Select[Range[nn],Length[Select[sqfics[#],UnsameQ@@hwt/@#&]]==0&]