%I #9 Jan 17 2020 10:37:38

%S 1,2,1,2,1,3,3,4,6,3,12,10,12,14,27,38,44,52,48,77,101,106,127,206,

%T 268,377,392,496,602,671,821,1090,1318,1568,1926,2260,2703,3258,3942,

%U 4858,5923,6891,8286,9728,11676,13775,16314,19749,23474,27793,32989,38775

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

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

%e The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):

%e 1 2 3 4 5 6 7 8 9 A B

%e 11 1111 222 3211 431 432 5311 542

%e 321 22111 4211 3321 22111111 5411

%e 11111111 32211 33221

%e 321111 42221

%e 2211111 53111

%e 322211

%e 431111

%e 521111

%e 2222111

%e 3311111

%e 32111111

%e For example, the partition (3,3,2,2,1) is counted under a(11) because 5*5*3*3*2 = 450 is divisible by 5+5+3+3+2 = 18.

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

%Y The Heinz numbers of these partitions are given by A036844.

%Y Numbers divisible by the sum of their prime indices are A324851.

%Y Partitions whose product is divisible by their sum are A057568.

%Y Partitions whose Heinz number is divisible by all parts are A330952.

%Y Partitions whose Heinz number is divisible by their product are A324925.

%Y Partitions whose Heinz number is divisible by their sum are A330950.

%Y Partitions whose product is divisible by their sum of primes are A330954.

%Y Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 15 2020