A381870 Numbers whose prime indices have a unique multiset partition into sets with distinct sums.

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Author

Gus Wiseman, Mar 12 2025

Keywords

• nonn

Comments

First differs from A212166 in lacking 360.

First differs from A293511 in having 600.

Also numbers with a unique factorization into squarefree numbers with distinct sums of prime indices (A056239).

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.

Examples

			For n = 600 the unique multiset partition is {{1},{1,3},{1,2,3}}. The unique factorization is 2*10*30.
		

Crossrefs

Without distinct block-sums we have A000961, ones in A050320.

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.

For distinct blocks instead of sums we have A293511, ones in A050326.

These are the positions of ones in A381633, see A381634, A381806, A381990.

Normal multiset partitions of this type are counted by A381718, see A279785.

For constant instead of strict blocks we have A381991, ones in A381635.

A001055 counts multiset partitions of prime indices, strict A045778.

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

A265947 counts refinement-ordered pairs of integer partitions.

A317141 counts coarsenings of prime indices, refinements A300383.

A321469 counts factorizations with distinct sums of prime indices, ones A166684.

Cf. A000720, A001222, A005117, A066328, A293243, A299202, A300385.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[Select[sfacs[#],UnsameQ@@hwt/@#&]]==1&]