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A072721
Number of partitions of n into parts which are each positive powers of a single number >1 (which may vary between partitions).
8
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.
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 *)
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jul 05 2002
STATUS
approved