A331381 - OEIS

%I #6 Jan 17 2020 10:37:53

%S 1,1,1,1,1,3,1,5,2,6,6,5,5,7,4,7,7,7,10,8,9,6,10,9,9,15,7,12,10,14,10,

%T 10,8,8,15,10,7,16,13,9,10,14,12,10,8,14,11,13,11,16,15,14,15,15,10,

%U 14,18,11,12,13,13,18,21,15,16,19,16,15,8,17,17

%N Number of integer partitions of n whose sum of primes of parts is divisible by their product of parts.

%e The a(n) partitions for n = 1, 5, 7, 8, 9, 13, 14:

%e   1  221    43       311111    63         7411           65111

%e      311    511      11111111  441        721111         322211111

%e      11111  3211               711        43111111       311111111111

%e             22111              42111      421111111      11111111111111

%e             1111111            2211111    3211111111

%e                                111111111  22111111111

%e                                           1111111111111

%t Table[Length[Select[IntegerPartitions[n],Divisible[Plus@@Prime/@#,Times@@#]&]],{n,0,30}]

%Y The Heinz numbers of these partitions are given by A331382.

%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 Sum of prime factors is divisible by sum of prime indices: A331380

%Y Partitions whose product is equal to their sum of primes are A331383.

%Y Product of prime indices equals sum of prime factors: A331384.

%Y Cf. A000040, A001414, A324850, A330953, A330954, A331378, A331379, A331415, A331416.

%K nonn

%O 0,6

%A _Gus Wiseman_, Jan 16 2020