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%I #10 Nov 29 2020 08:18:57
%S 1,0,1,1,1,1,2,1,3,2,4,3,5,4,6,6,7,7,10,8,11,12,12,14,17,16,20,22,24,
%T 26,31,31,37,39,43,46,54,53,63,65,73,75,87,87,100,102,115,117,133,134,
%U 151,155,172,176,197,202,223,231,254,262,290,298,327,341,370
%N Number of integer partitions of n where the multiplicity of each part k is at least prime(k).
%C The Heinz numbers of these partitions are given by A054744.
%F G.f.: Product_{k>=1} (1 + x^(prime(k)*k) / (1 - x^k)). - _Ilya Gutkovskiy_, Nov 28 2020
%e The first few terms count the following integer partitions:
%e 0: ()
%e 2: (11)
%e 3: (111)
%e 4: (1111)
%e 5: (11111)
%e 6: (222)
%e 6: (111111)
%e 7: (1111111)
%e 8: (2222)
%e 8: (22211)
%e 8: (11111111)
%e 9: (222111)
%e 9: (111111111)
%e 10: (22222)
%e 10: (222211)
%e 10: (2221111)
%e 10: (1111111111)
%e 11: (2222111)
%e 11: (22211111)
%e 11: (11111111111)
%e 12: (222222)
%e 12: (2222211)
%e 12: (22221111)
%e 12: (222111111)
%e 12: (111111111111)
%t Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=Prime[i],{i,Union[#]}]&]],{n,0,30}]
%Y Cf. A052335, A054744, A062457, A087153, A117144, A324525, A324572.
%K nonn
%O 0,7
%A _Gus Wiseman_, Apr 01 2019