%I #11 Jan 02 2019 11:40:19

%S 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,

%T 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,

%U 5,145,14,142,1,189,1,168,19,202,5,223,1,241,17,247,1,309,1,286,24,333,4

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

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

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

%F a(n) = A322900(n) - 1. - _Gus Wiseman_, Jan 01 2019

%e 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).

%e From _Gus Wiseman_, Jan 01 2019: (Start)

%e The a(2) = 1 through a(12) = 10 integer partitions (A = 10, B = 11, C = 12):

%e (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)

%e (22) (33) (44) (333) (55) (66)

%e (42) (422) (82) (84)

%e (222) (2222) (442) (93)

%e (4222) (444)

%e (22222) (822)

%e (3333)

%e (4422)

%e (42222)

%e (222222)

%e (End)

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

%t Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],SameQ@@radbase/@#]&]],{n,30}] (* _Gus Wiseman_, Jan 01 2019 *)

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

%K nonn

%O 0,5

%A _Henry Bottomley_, Jul 05 2002