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A072720 Number of partitions of n into parts which are each powers of a single number (which may vary between partitions). 8
1, 1, 2, 3, 5, 6, 10, 11, 15, 17, 23, 24, 34, 35, 43, 47, 57, 58, 73, 74, 91, 96, 112, 113, 139, 141, 163, 168, 197, 198, 235, 236, 272, 279, 317, 321, 378, 379, 427, 436, 501, 502, 575, 576, 653, 666, 742, 743, 851, 853, 952, 963, 1080, 1081, 1211, 1216, 1361 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

First differs from A322912 at a(12) = 34, A322912(12) = 33.

LINKS

Table of n, a(n) for n=0..56.

FORMULA

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

EXAMPLE

a(6)=10 since 6 can be written as 6 (powers of 6), 5+1 (5), 4+1+1 (4 or 2), 3+3 (3), 3+1+1+1 (3), 4+2 (2), 2+2+2 (2), 2+2+1+1 (2), 2+1+1+1+1 (2) and 1+1+1+1+1+1 (powers of anything).

From Gus Wiseman, Jan 01 2019: (Start)

The a(1) = 1 through a(8) = 15 integer partitions:

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

       (11)  (21)   (22)    (41)     (33)      (61)       (44)

             (111)  (31)    (221)    (42)      (331)      (71)

                    (211)   (311)    (51)      (421)      (422)

                    (1111)  (2111)   (222)     (511)      (611)

                            (11111)  (411)     (2221)     (2222)

                                     (2211)    (4111)     (3311)

                                     (3111)    (22111)    (4211)

                                     (21111)   (31111)    (5111)

                                     (111111)  (211111)   (22211)

                                               (1111111)  (41111)

                                                          (221111)

                                                          (311111)

                                                          (2111111)

                                                          (11111111)

(End)

MATHEMATICA

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

Table[Length[Select[IntegerPartitions[n], SameQ@@radbase/@DeleteCases[#, 1]&]], {n, 30}] (* Gus Wiseman, Jan 01 2019 *)

CROSSREFS

Cf. A000123, A005704, A005705, A005706, A072721.

Cf. A018819, A023893, A052410, A102430, A322900, A322901, A322902, A322903, A322912.

Sequence in context: A074243 A302539 A322912 * A298363 A018396 A003238

Adjacent sequences:  A072717 A072718 A072719 * A072721 A072722 A072723

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jul 05 2002

STATUS

approved

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Last modified February 16 12:48 EST 2019. Contains 320163 sequences. (Running on oeis4.)