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A340829
Number of strict integer partitions of n whose Heinz number (product of primes of parts) is divisible by n.
3
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
1,7
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.
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Divisible[Times@@Prime/@#, n]&]], {n, 30}]
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.
Sequence in context: A178516 A306418 A323886 * A352551 A174739 A280542
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 01 2021
STATUS
approved