A370595 Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.

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, 125, 148, 170, 190, 209, 250, 273, 305, 341, 403, 432, 484, 561, 623, 708, 765, 873, 977, 1109, 1178, 1367, 1493, 1669, 1824, 2054, 2265, 2521, 2770

Offset: 0

Author

Gus Wiseman, Mar 03 2024

Keywords

• nonn

Comments

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

Examples

			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
		

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.

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

Programs

Mathematica

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

Extensions

More terms from Jinyuan Wang, Feb 14 2025