A387181 Heinz numbers of integer partitions with no part k appearing more than A000005(k) times.

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, 94, 95, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 122, 123, 126

Offset

1,2

Comments

First differs from A276078 and A048103 in lacking 125.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Also numbers of which it is possible to choose a different constant integer partition of each prime index. The multiset of prime indices of n is row n of A112798.

Links

Table of n, a(n) for n=1..91.

Example

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 not 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 in the sequence.

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@@#&]!={}&]
- or -
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], And@@Table[Count[prix[#], k]<=DivisorSigma[0, k], {k, Union[prix[#]]}]&]

Crossrefs

The complement for divisors is A355740, counted by A370320.

The complement for prime factors is A355529, counted by A370593.

For prime factors instead of constant partitions we have A368100, counted by A370592.

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

These are all positions of nonzero terms in A387120.

The complement for strict partitions is A387176, counted by A387137.

For strict instead of constant partitions we have A387177, counted by A387178.

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

The complement is A387180, counted by A387329.

Partitions of this type are counted by A387330.

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. A000005, A052335, A063834, A238873, A276079, A335433, A335448, A355739, A383706, A387110.

Sequence in context: A390361 A048103 A276078 * A193303 A285465 A092418

Adjacent sequences: A387178 A387179 A387180 * A387182 A387183 A387184

Keyword

nonn

Author

Gus Wiseman, Sep 04 2025

Status

approved