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

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, 56, 0, 64, 66, 86, 0, 125, 104, 135, 0, 196, 111, 230, 0, 0, 274, 353, 0, 0, 0, 563, 0, 687, 0, 974, 0, 1039, 1052, 1290, 0, 1473, 1511, 0, 0, 2707, 1614, 2664, 0

Offset: 1

Author

Gus Wiseman, Feb 01 2021

Keywords

• nonn

Comments

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.

Examples

			The a(6) = 1 through a(19) = 10 partitions (empty columns indicated by dots, A = 10, B = 11):
  321  43   .  .  631   65    .  76    941   A32    .  A7     .  B8
       421        4321  542      643   6431  6432      764       865
                        5321     652   7421  9321      872       874
                                 6421        54321     971       982
                                                       7532      A81
                                                       7541      8542
                                                       7631      8632
                                                       74321     8641
                                                                 8731
                                                                 85321
		

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

Positions of zeros are 2 and A013929.

The non-strict version is A330950 (A324851) q.v.

A000009 counts strict partitions.

A003963 multiplies together prime indices.

A018818 counts partitions into divisors (A326841).

A047993 counts balanced partitions (A106529).

A056239 adds up prime indices.

A057568 counts partitions whose product is divisible by their sum (A326149).

A067538 counts partitions whose length/max divides sum (A316413/A326836).

A072233 counts partitions by sum and length, with strict case A008289.

A102627 counts strict partitions whose length divides sum.

A112798 lists the prime indices of each positive integer.

A120383 lists numbers divisible by all of their prime indices.

A324850 lists numbers divisible by the product of their prime indices.

A324925 counts partitions whose Heinz number is divisible by their product.

A326842 counts partitions whose parts and length all divide sum (A326847).

A326850 counts strict partitions whose maximum part divides sum.

A326851 counts strict partitions with length and maximum dividing sum.

A330952 counts partitions whose Heinz number is divisible by all parts.

A340828 counts strict partitions with length divisible by maximum.

A340830 counts strict partitions with parts divisible by length.

Cf. A052335, A064173, A064174, A096401, A143773 (A316428), A168659 (A340609/A340610), A326843 (A326837).

Programs

Mathematica

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