A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 0, 1, 1, 0, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1

Offset

1,8

Comments

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

Links

Robert Price, Table of n, a(n) for n = 1..1000

Example

The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.

Mathematica

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

Crossrefs

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).

For distinct blocks instead of sums we have A050361, after sums A381715.

For strict instead of constant we have A381633 (zeros A381806), after sums A381634.

Positions of 0 are A381636.

Taking block-sums (and sorting) gives A381716.

Other multiset partitions of prime indices:

More on multiset partitions into constant blocks: A006171, A279784, A295935.

A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A000720, A001222, A002846, A005117, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

Sequence in context: A323191 A257179 A259042 * A381716 A350074 A333179

Adjacent sequences: A381632 A381633 A381634 * A381636 A381637 A381638

Keyword

nonn

Author

Gus Wiseman, Mar 09 2025

Status

approved