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

Author

Gus Wiseman, May 16 2022

Keywords

• nonn
• more

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).

Examples

			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)
		

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.

Programs

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}]