A387576 Numbers k such that it is possible to choose a sequence of distinct integer partitions, one of each prime index of k (with multiplicity).

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 90, 91, 93

Offset

1,2

Comments

First differs from A276078 in having 16807.

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.

Also numbers with no prime factor prime(x) whose exponent exceeds A000041(x).

Where does this first differ from A387181 and A048103? - R. J. Mathar, Sep 17 2025

Links

Table of n, a(n) for n=1..67.

Example

The prime indices of 125 are (3,3,3), and we have choices such as ((3),(1,1,1),(2,1)), so 125 is included.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   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}

Mathematica

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

Crossrefs

For initial intervals instead of partitions we have A276078, complement A276079.

For divisors instead of partitions we have A368110, counted by A239312.

These are positions of nonzero terms in A387115.

For sets of strict partitions we have A387177, complement A387176.

Integer partitions of this type are counted by A387328.

The complement is A387577, counted by A387134.

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.

A238873 counts superdiagonal partitions, ranks A387112.

A239312 counts partitions with choosable divisors, strict A387178.

A289509 lists numbers with relatively prime prime indices.

A370320 counts partitions without choosable divisors, strict A387137.

A387118 counts non-superdiagonal partitions, ranks A387113.

Cf. A000720, A052335, A052337, A120383, A335433, A335448, A355740, A367771, A383706, A387110, A387181.

Sequence in context: A195291 A042968 A337037 * A390361 A048103 A276078

Adjacent sequences: A387573 A387574 A387575 * A387577 A387578 A387579

Keyword

nonn

Author

Gus Wiseman, Sep 07 2025

Status

approved