A353426 Number of integer partitions of n that are empty or a singleton or whose multiplicities are a sub-multiset that is already counted.

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 3, 3, 5, 4, 6, 5, 6, 6, 7, 8, 10, 12, 12, 14, 13, 13, 18, 15, 16, 19, 20, 20, 32, 37, 53, 74, 105

Offset

0,5

Comments

a(n) is number of integer partitions of n whose Heinz number belongs to A353393, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Links

Table of n, a(n) for n=0..40.

Example

The a(n) partitions for selected n (A..M = 10..22):
  n=1: n=4:  n=14:     n=16:     n=17:     n=18:        n=22:
------------------------------------------------------------------
  (1)  (4)   (E)       (G)       (H)       (I)          (M)
       (22)  (5522)    (4444)    (652211)  (7722)       (9922)
             (532211)  (6622)    (742211)  (752211)     (972211)
                       (642211)  (832211)  (842211)     (A62211)
                       (732211)            (932211)     (B52211)
                                           (333222111)  (C42211)
                                                        (D32211)

Mathematica

oosQ[y_]:=Length[y]<=1||MemberQ[Subsets[Sort[y], {Length[Union[y]]}], Sort[Length/@Split[y]]]&&oosQ[Sort[Length/@Split[y]]];
Table[Length[Select[IntegerPartitions[n], oosQ]], {n, 0, 30}]

Crossrefs

The non-recursive version is A325702, ranked by A325755.

The version for compositions is A353391, non-recursive A353390.

These partitions are ranked by A353393, nonprime A353389.

A047966 counts uniform partitions, compositions A329738.

A239455 counts Look-and-Say partitions, ranked by A351294.

Cf. A000041, A002033, A074761, A181819, A181821, A323014, A325534, A333755.

Sequence in context: A139355 A039736 A093921 * A140192 A324905 A240231

Adjacent sequences: A353423 A353424 A353425 * A353427 A353428 A353429

Keyword

nonn,more

Author

Gus Wiseman, May 16 2022

Status

approved