A370595 - OEIS

%I #11 Feb 14 2025 09:46:00

%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 93,94,112,125,148,170,190,209,250,273,305,341,403,432,484,561,623,

%U 708,765,873,977,1109,1178,1367,1493,1669,1824,2054,2265,2521,2770

%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,changed

%O 0,5

%A _Gus Wiseman_, Mar 03 2024

%E More terms from _Jinyuan Wang_, Feb 14 2025