A381870 - OEIS

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..61.

EXAMPLE

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

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&]

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.

Sequence in context: A336418 A212166 A293511 * A384084 A336615 A385576

Adjacent sequences: A381867 A381868 A381869 * A381871 A381872 A381873

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 12 2025

STATUS

approved