A370595 - OEIS

A370595

Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

%I #7 May 24 2024 03:50:46

%S 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,

%T 93,94,112

%N Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

%C For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.

%e The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):

%e 1 . 21 22 . 33 322 71 441 55 533 B1 553 77 933

%e 31 51 421 332 522 442 722 444 733 D1 B22

%e 321 422 531 721 731 552 751 B21 B31

%e 521 4321 4322 4332 931 4433 4443

%e 5321 4431 4432 5441 5442

%e 5322 5332 6332 5532

%e 5421 5422 7322 6621

%e 6321 6322 7421 7332

%e 7321 7422

%e 7521

%e 8421

%e 9321

%e 54321

%t Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]

%Y For no choices we have A370320, complement A239312.

%Y The version for prime factors (not all divisors) is A370594, ranks A370647.

%Y For multiple choices we have A370803, ranks A370811.

%Y These partitions have ranks A370810.

%Y A000005 counts divisors.

%Y A000041 counts integer partitions, strict A000009.

%Y A027746 lists prime factors, A112798 indices, length A001222.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y A355741, A355744, A355745 choose prime factors of prime indices.

%Y A370592 counts partitions with choosable prime factors, ranks A368100.

%Y A370593 counts partitions without choosable prime factors, ranks A355529.

%Y A370804 counts non-condensed partitions with no ones, complement A370805.

%Y A370814 counts factorizations with choosable divisors, complement A370813.

%Y Cf. A355535, A355739, A355740, A367867, A367904, A368110, A370583, A370584, A370806, A370808.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Mar 03 2024