A387176 Numbers whose prime indices do not have choosable sets of strict integer partitions. Zeros of A387115.

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 52, 54, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 90, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 162, 164, 168, 171, 172

Offset: 1

Author

Gus Wiseman, Aug 27 2025

Keywords

• nonn
• new

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.

Crossrefs

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

For all partitions we appear to have A276079, counted by A387134.

For divisors instead of strict partitions we have A355740, counted by A370320.

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

The complement for divisors is A368110, counted by A239312.

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

For initial intervals instead of strict partitions we have A387113, counted by A387118.

These are the positions of 0 in A387115.

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

The complement is A387177.

The version for constant partitions is A387180, counted by A387329.

The complement 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.

A120383 lists numbers divisible by all of their prime indices.

A289509 lists numbers with relatively prime prime indices.

Cf. A000720, A261049, A270995, A335433, A335448, A355744, A357978, A357980, A383706, A387110, A387120.

Programs

Mathematica

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