A330950 - OEIS

%I #5 Jan 15 2020 18:39:36

%S 1,1,1,2,2,3,3,7,7,11,11,22,15,30,42,77,42,101,56,176,176,231,135,490,

%T 490,490,792,1002,490,1575,627,3010,2436,2436,3718,5604,1958,4565,

%U 6842,12310,3718,14883,4565,21637,26015,17977,8349,53174,44583,63261

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

%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(10) = 11 partitions:

%e   1  11  21  211   32   321    43    5111      522      631

%e              1111  311  2211   421   32111     3222     3331

%e                         21111  4111  41111     4221     4321

%e                                      221111    22221    5311

%e                                      311111    32211    32221

%e                                      2111111   222111   33211

%e                                      11111111  2211111  43111

%e                                                         322111

%e                                                         331111

%e                                                         3211111

%e                                                         31111111

%e For example, the Heinz number of (3,2) is 15, which is divisible by 5, so (3,2) is counted under a(5).

%t Table[Length[Select[IntegerPartitions[n],Divisible[Times@@Prime/@#,n]&]],{n,20}]

%Y The Heinz numbers of these partitions are given by 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 Cf. A056239, A112798, A196050, A324850, A324924, A330953, A330954, A331379, A331381, A331383, A331384.

%K nonn

%O 1,4

%A _Gus Wiseman_, Jan 15 2020