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A322900 Number of integer partitions of n whose parts are all proper powers of the same number. 7
1, 1, 2, 2, 3, 2, 5, 2, 5, 3, 7, 2, 11, 2, 9, 5, 11, 2, 16, 2, 18, 6, 17, 2, 27, 3, 23, 6, 30, 2, 38, 2, 37, 8, 39, 5, 58, 2, 49, 10, 66, 2, 74, 2, 78, 14, 77, 2, 109, 3, 100, 12, 118, 2, 131, 6, 146, 15, 143, 2, 190, 2, 169, 20, 203, 6, 224, 2, 242, 18, 248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Such a partition contains either no 1's or only 1's.

A proper power of n is a number n^k for some positive integer k.

Also integer partitions whose parts all have the same radical base (A052410).

LINKS

Table of n, a(n) for n=0..70.

EXAMPLE

The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):

  (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)

      (11) (111) (22)   (11111) (33)     (1111111) (44)       (333)

                 (1111)         (42)               (422)      (111111111)

                                (222)              (2222)

                                (111111)           (11111111)

.

  (A)          (B)           (C)            (D)             (E)

  (55)         (11111111111) (66)           (1111111111111) (77)

  (82)                       (84)                           (842)

  (442)                      (93)                           (4442)

  (4222)                     (444)                          (8222)

  (22222)                    (822)                          (44222)

  (1111111111)               (3333)                         (422222)

                             (4422)                         (2222222)

                             (42222)                        (11111111111111)

                             (222222)

                             (111111111111)

MATHEMATICA

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All, 2]]);

Table[Length[Select[IntegerPartitions[n], SameQ@@radbase/@#&]], {n, 30}]

CROSSREFS

a(n) = A072721(n) + 1.

Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.

Sequence in context: A180125 A272209 A323347 * A238791 A007012 A298423

Adjacent sequences:  A322897 A322898 A322899 * A322901 A322902 A322903

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 30 2018

STATUS

approved

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Last modified April 18 12:34 EDT 2019. Contains 322209 sequences. (Running on oeis4.)