cp's OEIS Frontend

A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.

%I A340829 #10 Feb 03 2021 09:08:52
%S A340829 1,0,1,0,1,1,2,0,0,2,3,0,4,3,4,0,8,0,10,0,11,12,19,0,0,22,0,0,46,23,
%T A340829 56,0,64,66,86,0,125,104,135,0,196,111,230,0,0,274,353,0,0,0,563,0,
%U A340829 687,0,974,0,1039,1052,1290,0,1473,1511,0,0,2707,1614,2664,0
%N A340829 Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.
%C A340829 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. The Heinz numbers of these partitions are squarefree numbers divisible by the sum of their prime indices.
%e A340829 The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
%e A340829   321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
%e A340829        421        4321  542      643   6431  6432      764       865
%e A340829                         5321     652   7421  9321      872       874
%e A340829                                  6421        54321     971       982
%e A340829                                                        7532      A81
%e A340829                                                        7541      8542
%e A340829                                                        7631      8632
%e A340829                                                        74321     8641
%e A340829                                                                  8731
%e A340829                                                                  85321
%t A340829 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@Prime/@#,n]&]],{n,30}]
%Y A340829 Note: A-numbers of Heinz-number sequences are in parentheses below.
%Y A340829 Positions of zeros are 2 and A013929.
%Y A340829 The non-strict version is A330950 (A324851) q.v.
%Y A340829 A000009 counts strict partitions.
%Y A340829 A003963 multiplies together prime indices.
%Y A340829 A018818 counts partitions into divisors (A326841).
%Y A340829 A047993 counts balanced partitions (A106529).
%Y A340829 A056239 adds up prime indices.
%Y A340829 A057568 counts partitions whose product is divisible by their sum (A326149).
%Y A340829 A067538 counts partitions whose length/max divides sum (A316413/A326836).
%Y A340829 A072233 counts partitions by sum and length, with strict case A008289.
%Y A340829 A102627 counts strict partitions whose length divides sum.
%Y A340829 A112798 lists the prime indices of each positive integer.
%Y A340829 A120383 lists numbers divisible by all of their prime indices.
%Y A340829 A324850 lists numbers divisible by the product of their prime indices.
%Y A340829 A324925 counts partitions whose Heinz number is divisible by their product.
%Y A340829 A326842 counts partitions whose parts and length all divide sum (A326847).
%Y A340829 A326850 counts strict partitions whose maximum part divides sum.
%Y A340829 A326851 counts strict partitions with length and maximum dividing sum.
%Y A340829 A330952 counts partitions whose Heinz number is divisible by all parts.
%Y A340829 A340828 counts strict partitions with length divisible by maximum.
%Y A340829 A340830 counts strict partitions with parts divisible by length.
%Y A340829 Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).
%K A340829 nonn
%O A340829 1,7
%A A340829 _Gus Wiseman_, Feb 01 2021