A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset

1,2

Comments

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

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

Robert Price, Table of n, a(n) for n = 1..1247

Example

The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

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]}]];
Select[Range[100], Length[Select[pfacs[#], UnsameQ@@hwt/@#&]]==1&]

Crossrefs

For distinct blocks instead of block-sums we have A004709, counted by A000726.

Twice-partitions of this type are counted by A279786.

MM-numbers of these multiset partitions are A326535 /\ A355743.

These are the positions of 1 in A381635.

For no choices we have A381636 (zeros of A381635), counted by A381717.

For strict instead of constant blocks we have A381870, counted by A382079.

Partitions of this type (unique into constant with distinct) are counted by A382301.

Normal multiset partitions of this type are counted by A382203.

A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.

Cf. A006171, A279784, A295935.

Sequence in context: A210490 A340682 A166155 * A394994 A392698 A342525

Adjacent sequences: A381988 A381989 A381990 * A381992 A381993 A381994

Keyword

nonn

Author

Gus Wiseman, Mar 22 2025

Status

approved