A387180 Numbers of which it is not possible to choose a different constant integer partition of each prime index.

4, 8, 12, 16, 20, 24, 27, 28, 32, 36, 40, 44, 48, 52, 54, 56, 60, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 156, 160, 162, 164, 168, 172, 176, 180, 184, 188, 189, 192, 196, 200, 204

Offset: 1

Author

Gus Wiseman, Aug 30 2025

Keywords

• nonn
• new

Comments

First differs from A276079 in having 125.

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 n with at least one prime index k such that the multiplicity of prime(k) in the prime factorization of n exceeds the number of divisors of k.

Examples

			The prime indices of 60 are {1,1,2,3}, and we have the following 4 choices of constant partitions:
  ((1),(1),(2),(3))
  ((1),(1),(2),(1,1,1))
  ((1),(1),(1,1),(3))
  ((1),(1),(1,1),(1,1,1))
Since none of these is strict, 60 is in the sequence.
The prime indices of 90 are {1,2,2,3}, and we have the following 4 strict choices:
  ((1),(2),(1,1),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(3))
  ((1),(1,1),(2),(1,1,1))
So 90 is not in the sequence.
		

Crossrefs

For prime factors instead of constant partitions we have A355529, counted by A370593.

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

The complement for prime factors is A368100, counted by A370592.

The complement for divisors is A368110, counted by A239312.

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

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

These are the positions of zero in A387120.

For strict instead of constant partitions we have A387176, counted by A387137.

The complement for strict partitions is A387177, counted by A387178.

Twice-partitions of this type are counted by A387179, constant-block case of A296122.

The complement is A387181 (nonzeros of A387120), counted by A387330.

Partitions of this type are counted by A387329.

A000041 counts integer partitions, strict A000009.

A003963 multiplies together prime indices.

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. A355739, A367771, A387111, A387115.

Cf. A000005, A052335, A063834, A276079, A299200, A299201, A335433, A335448, A355731, A383706, A387110.

Programs

Mathematica

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