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A371172
Number of integer partitions of n with as many submultisets as distinct divisors of parts.
12
0, 0, 1, 1, 0, 1, 0, 3, 2, 3, 1, 4, 2, 1, 2, 3, 4, 2, 4, 1, 5, 2, 7, 5, 9, 4, 9, 15, 18, 16, 24, 13, 17, 23, 23, 22, 34, 17, 30, 31, 36, 29, 43, 21, 30, 35, 44, 28, 47, 19, 44
OFFSET
0,8
COMMENTS
The Heinz numbers of these partitions are given by A371165.
EXAMPLE
The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
The a(17) = 2 through a(24) = 9 partitions:
(17) (9,9) (19) (11,9) (14,7) (13,9) (23) (21,3)
(13,4) (15,3) (15,5) (17,4) (21,1) (19,4) (22,2)
(6,6,6) (8,6,6) (8,8,6) (22,1) (8,8,8)
(12,3,3) (12,4,4) (10,6,6) (15,4,4) (10,8,6)
(18,1,1) (16,3,3) (12,10,1) (12,6,6)
(18,2,2) (12,7,5)
(20,1,1) (18,3,3)
(20,2,2)
(12,10,2)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]], {n, 0, 30}]
CROSSREFS
The RHS is represented by A370820.
Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.
These partitions are ranked by A371165.
A000005 counts divisors.
A355731 counts choices of a divisor of each prime index, firsts A355732.
Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
Sequence in context: A213940 A236027 A220128 * A324468 A131498 A236455
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 16 2024
STATUS
approved