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%I #9 Apr 10 2020 15:04:35
%S 0,0,0,0,0,0,1,0,2,2,1,1,1,2,2,2,1,4,2,2,2,4,2,3,4,1,3,4,5,0,3,3,1,6,
%T 2,1,5,4,2,3,4,2,2,3,1,5,2,3,4,6,5,2,7,1,3,5,3,4,2,5,5,4,7,3,6,4,4,2,
%U 4,4,3,9,4,3,5,3,5,4,4,4,3,7,4,2,8,2,3
%N Number of integer partitions of n whose sum of primes of parts is equal to their product of parts.
%e The a(n) partitions for n = 7, 9, 18, 24:
%e (4,3) (6,3) (12,4,1,1) (19,4,1)
%e (4,4,1) (11,4,1,1,1) (18,4,1,1)
%e (8,5,1,1,1,1,1) (9,6,1,1,1,1,1,1,1,1,1)
%e (4,2,2,2,1,1,1,1,1,1,1,1)
%e For example, (4,4,1) has sum of primes of parts 7+7+2 = 16 and product of parts 4*4*1 = 16, so is counted under a(9).
%t Table[Length[Select[IntegerPartitions[n],Times@@#==Plus@@Prime/@#&]],{n,30}]
%Y The Heinz numbers of these partitions are given by A331384.
%Y Numbers divisible by the sum of their prime factors are A036844.
%Y Partitions whose product is divisible by their sum are A057568.
%Y Numbers divisible by the sum of their prime indices are A324851.
%Y Product of prime indices is divisible by sum of prime indices: A326149.
%Y Partitions whose Heinz number is divisible by their sum are A330950.
%Y Partitions whose Heinz number is divisible by their sum of primes: A330953.
%Y Sum of prime factors is divisible by sum of prime indices: A331380
%Y Partitions whose product divides their sum of primes are A331381.
%Y Cf. A000040, A001414, A324850, A330954, A331378, A331379, A331382, A331415, A331416.
%K nonn,more
%O 1,9
%A _Gus Wiseman_, Jan 16 2020
%E a(71)-a(87) from _Robert Price_, Apr 10 2020
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