cp's OEIS Frontend

A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.

%I A330953 #9 Jan 17 2020 10:37:38
%S A330953 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 A330953 268,377,392,496,602,671,821,1090,1318,1568,1926,2260,2703,3258,3942,
%U A330953 4858,5923,6891,8286,9728,11676,13775,16314,19749,23474,27793,32989,38775
%N A330953 Number of integer partitions of n whose Heinz number (product of primes of parts) is divisible by their sum of primes of parts.
%C A330953 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 A330953 The a(1) = 1 through a(11) = 12 partitions: (A = 10, B = 11):
%e A330953   1  2   3  4     5  6    7      8         9        A         B
%e A330953      11     1111     222  3211   431       432      5311      542
%e A330953                      321  22111  4211      3321     22111111  5411
%e A330953                                  11111111  32211              33221
%e A330953                                            321111             42221
%e A330953                                            2211111            53111
%e A330953                                                               322211
%e A330953                                                               431111
%e A330953                                                               521111
%e A330953                                                               2222111
%e A330953                                                               3311111
%e A330953                                                               32111111
%e A330953 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 A330953 Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,Plus@@Prime/@#]&]],{n,30}]
%Y A330953 The Heinz numbers of these partitions are given by A036844.
%Y A330953 Numbers divisible by the sum of their prime indices are A324851.
%Y A330953 Partitions whose product is divisible by their sum are A057568.
%Y A330953 Partitions whose Heinz number is divisible by all parts are A330952.
%Y A330953 Partitions whose Heinz number is divisible by their product are A324925.
%Y A330953 Partitions whose Heinz number is divisible by their sum are A330950.
%Y A330953 Partitions whose product is divisible by their sum of primes are A330954.
%Y A330953 Cf. A001414, A003963, A056239, A112798, A120383, A326149, A326155, A331378, A331379, A331381, A331383, A331415, A331416, A331417.
%K A330953 nonn
%O A330953 1,2
%A A330953 _Gus Wiseman_, Jan 15 2020