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A322900
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Number of integer partitions of n whose parts are all proper powers of the same number.
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7
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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
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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)
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(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)
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MATHEMATICA
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radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All, 2]]);
Table[Length[Select[IntegerPartitions[n], SameQ@@radbase/@#&]], {n, 30}]
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CROSSREFS
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Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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