A072721 Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 10, 1, 8, 4, 10, 1, 15, 1, 17, 5, 16, 1, 26, 2, 22, 5, 29, 1, 37, 1, 36, 7, 38, 4, 57, 1, 48, 9, 65, 1, 73, 1, 77, 13, 76, 1, 108, 2, 99, 11, 117, 1, 130, 5, 145, 14, 142, 1, 189, 1, 168, 19, 202, 5, 223, 1, 241, 17, 247, 1, 309, 1, 286, 24, 333, 4

Offset

0,5

Comments

First differs from A322968 at a(12) = 10, A322968(12) = 9.

Links

Table of n, a(n) for n=0..77.

Formula

a(n) = A072721(n)-A072721(n-1). a(p)=1 for p prime.

a(n) = A322900(n) - 1. - Gus Wiseman, Jan 01 2019

Example

a(5)=1 since the only partition without 1 as a part is 5 (a power of 5). a(6)=4 since 6 can be written as 6 (powers of 6), 3+3 (powers of 3) and 4+2 and 2+2+2 (both powers of 2).
From Gus Wiseman, Jan 01 2019: (Start)
The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (C)
            (22)       (33)        (44)    (333)  (55)          (66)
                       (42)        (422)          (82)          (84)
                       (222)       (2222)         (442)         (93)
                                                  (4222)        (444)
                                                  (22222)       (822)
                                                                (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
(End)

Mathematica

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All, 2]]);
Table[Length[Select[IntegerPartitions[n], And[FreeQ[#, 1], SameQ@@radbase/@#]&]], {n, 30}] (* Gus Wiseman, Jan 01 2019 *)

Crossrefs

Cf. A018819, A023894, A052410, A072720, A072721, A102430, A322900, A322902, A322903, A322968.

Sequence in context: A029205 A229340 A322968 * A339792 A285711 A035092

Adjacent sequences: A072718 A072719 A072720 * A072722 A072723 A072724

Keyword

nonn

Author

Henry Bottomley, Jul 05 2002

Status

approved