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%I #6 Jul 29 2022 09:51:27
%S 0,0,0,0,0,1,0,3,2,5,4,11,7,18,16,26,27,43,41,65,65,92,100,137,142,
%T 194,210,270,295,379,410,519,571,699,782,947,1046,1267,1414,1673,1870,
%U 2213,2465,2897,3230,3757,4210,4871,5427,6265,6997
%N Number of integer partitions of n into relatively prime prime-powers.
%e The a(5) = 1 through a(12) = 7 partitions:
%e (32) . (43) (53) (54) (73) (74) (75)
%e (52) (332) (72) (433) (83) (543)
%e (322) (432) (532) (92) (552)
%e (522) (3322) (443) (732)
%e (3222) (533) (4332)
%e (542) (5322)
%e (722) (33222)
%e (3332)
%e (4322)
%e (5222)
%e (32222)
%t Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&&GCD@@#==1&]],{n,0,30}]
%Y This is the relatively prime case of A023894, facs A000688, w/ 1's A023893.
%Y For strict instead of coprime: A054685, facs A050361, with 1's A106244.
%Y The version for factorizations instead of partitions is A354911.
%Y A000041 counts partitions, strict A000009.
%Y A072233 counts partitions by sum and length.
%Y A246655 lists the prime-powers (A000961 includes 1), towers A164336.
%Y A279784 counts twice-partitions where the latter partitions are constant.
%Y A289509 lists numbers whose prime indices are relatively prime.
%Y A355743 lists numbers with prime-power prime indices, squarefree A356065.
%Y Cf. A001970, A055887, A063834, A076610, A085970, A355737, A355742.
%K nonn
%O 0,8
%A _Gus Wiseman_, Jul 28 2022