A387177 Numbers whose prime indices have choosable sets of strict integer partitions. Positions of nonzero terms in A387115.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset

1,2

Comments

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.

We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.

Links

Table of n, a(n) for n=1..66.

Example

The prime indices of 50 are {1,3,3}, and {(1),(3),(2,1)} is a valid choice of distinct strict partitions, so 50 is in the sequence.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
strptns[n_]:=Select[IntegerPartitions[n], UnsameQ@@#&];
Select[Range[100], Select[Tuples[strptns/@prix[#]], UnsameQ@@#&]!={}&]

Crossrefs

The version for all partitions appears to be A276078, counted by A052335.

The complement for all partitions appears to be A276079, counted by A387134.

The complement for divisors is A355740, counted by A370320.

Twice-partitions of this type (into distinct strict partitions) are counted by A358914.

The version for divisors is A368110, counted by A239312.

The version for initial intervals is A387112, counted by A238873, see A387111.

The complement for initial intervals is A387113, counted by A387118.

These are the positions of nonzero terms in A387115.

The complement is A387176.

Partitions of this type are counted by A387178, complement A387137.

The complement for constant partitions is A387180, counted by A387329, see A387120.

The version for constant partitions is A387181, counted by A387330.

A000041 counts integer partitions, strict A000009.

A003963 multiplies together the prime indices of n.

A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.

A289509 lists numbers with relatively prime prime indices.

Cf. A000720, A120383, A270995, A299200, A335433, A335448, A357978, A357982, A383706, A387110.

Sequence in context: A358976 A064594 A325511 * A382913 A240370 A193304

Adjacent sequences: A387174 A387175 A387176 * A387178 A387179 A387180

Keyword

nonn

Author

Gus Wiseman, Aug 29 2025

Status

approved