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A370595
Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.
14
1, 1, 0, 1, 2, 0, 3, 2, 4, 3, 4, 5, 8, 9, 8, 13, 12, 17, 16, 27, 28, 33, 36, 39, 50, 58, 65, 75, 93, 94, 112
OFFSET
0,5
COMMENTS
For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.
EXAMPLE
The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
1 . 21 22 . 33 322 71 441 55 533 B1 553 77 933
31 51 421 332 522 442 722 444 733 D1 B22
321 422 531 721 731 552 751 B21 B31
521 4321 4322 4332 931 4433 4443
5321 4431 4432 5441 5442
5322 5332 6332 5532
5421 5422 7322 6621
6321 6322 7421 7332
7321 7422
7521
8421
9321
54321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[Sort /@ Select[Tuples[Divisors/@#], UnsameQ@@#&]]]==1&]], {n, 0, 30}]
CROSSREFS
For no choices we have A370320, complement A239312.
The version for prime factors (not all divisors) is A370594, ranks A370647.
For multiple choices we have A370803, ranks A370811.
These partitions have ranks A370810.
A000005 counts divisors.
A000041 counts integer partitions, strict A000009.
A027746 lists prime factors, A112798 indices, length A001222.
A355731 counts choices of a divisor of each prime index, firsts A355732.
A355741, A355744, A355745 choose prime factors of prime indices.
A370592 counts partitions with choosable prime factors, ranks A368100.
A370593 counts partitions without choosable prime factors, ranks A355529.
A370804 counts non-condensed partitions with no ones, complement A370805.
A370814 counts factorizations with choosable divisors, complement A370813.
Sequence in context: A357866 A331622 A212175 * A289441 A291272 A291273
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 03 2024
STATUS
approved