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%I #11 Oct 12 2018 00:51:54
%S 0,1,2,4,5,8,9,13,15,20,21,29,30,37,40,50,51,64,65,80,84,99,100,123,
%T 125,146,151,178,179,212,213,249,255,292,295,348,349,396,404,466,467,
%U 535,536,611,622,697,698,801,803,900,910,1025,1026,1152,1156,1298,1311
%N Number of partitions of n into parts which are all powers of the same prime.
%C The exponents cannot all be zero.
%H Andrew Howroyd, <a href="/A064573/b064573.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{k>=1} 1/(Product_{r>=0} 1-x^(prime(k)^r)) - 1/(1-x). - _Andrew Howroyd_, Dec 29 2017
%e a(5)=5: 5^1, 3^1+2*3^0, 2^2+1, 2*2^1+1, 2^1+3*2^0
%e From _Gus Wiseman_, Oct 10 2018: (Start)
%e The a(2) = 1 through a(9) = 15 integer partitions:
%e (2) (3) (4) (5) (33) (7) (8) (9)
%e (21) (22) (41) (42) (331) (44) (81)
%e (31) (221) (51) (421) (71) (333)
%e (211) (311) (222) (511) (422) (441)
%e (2111) (411) (2221) (2222) (711)
%e (2211) (4111) (3311) (4221)
%e (3111) (22111) (4211) (22221)
%e (21111) (31111) (5111) (33111)
%e (211111) (22211) (42111)
%e (41111) (51111)
%e (221111) (222111)
%e (311111) (411111)
%e (2111111) (2211111)
%e (3111111)
%e (21111111)
%e (End)
%t Table[Length[Select[IntegerPartitions[n],PrimePowerQ[Times@@#]&]],{n,30}] (* _Gus Wiseman_, Oct 10 2018 *)
%o (PARI) first(n)={Vec(sum(k=2, n, if(isprime(k), 1/prod(r=0, logint(n,k), 1-x^(k^r) + O(x*x^n)) - 1/(1-x), 0)), -n)} \\ _Andrew Howroyd_, Dec 29 2017
%Y Cf. A028422, A064572, A064574, A064575, A064576, A064577, A319071, A320322, A320325.
%K easy,nonn
%O 1,3
%A _Marc LeBrun_, Sep 20 2001
%E Name clarified by _Andrew Howroyd_, Dec 29 2017
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