login
A370803
Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.
19
0, 0, 1, 1, 1, 3, 2, 4, 5, 7, 10, 11, 15, 18, 25, 28, 39, 45, 59, 66, 83, 101, 123, 150, 176, 213, 252, 301, 352, 426, 497, 589, 684, 802, 939, 1095, 1270
OFFSET
0,6
EXAMPLE
The partition (6,4,4,1) has two choices, namely {1,2,4,6} and {1,2,3,4}, so is counted under a(15).
The a(0) = 0 through a(13) = 18 partitions (A..D = 10..13):
. . 2 3 4 5 6 7 8 9 A B C D
32 42 43 44 54 64 65 66 76
41 52 53 63 73 74 75 85
61 62 72 82 83 84 94
431 81 91 92 93 A3
432 433 A1 A2 B2
621 532 443 543 C1
541 542 633 544
622 632 642 643
631 641 651 652
821 732 661
741 742
822 832
831 841
921 922
A21
5431
6421
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[Sort /@ Select[Tuples[Divisors/@#], UnsameQ@@#&]]]>1&]], {n, 0, 30}]
CROSSREFS
Including partitions with one choice gives A239312, complement A370320.
For a unique choice we have A370595, ranks A370810.
These partitions have ranks A370811.
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.
A355733 counts divisor-choices of prime indices.
A355741, A355744, A355745 choose prime factors of prime indices.
A370592 counts factor-choosable partitions, ranks A368100.
A370593 counts non-factor-choosable partitions, ranks A355529.
Sequence in context: A345866 A125060 A039882 * A326357 A354731 A164287
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Mar 03 2024
STATUS
approved