This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370595 #11 Feb 14 2025 09:46:00
%S A370595 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 A370595 93,94,112,125,148,170,190,209,250,273,305,341,403,432,484,561,623,
%U A370595 708,765,873,977,1109,1178,1367,1493,1669,1824,2054,2265,2521,2770
%N A370595 Number of integer partitions of n such that only one set can be obtained by choosing a different divisor of each part.
%C A370595 For example, the only choice for the partition (9,9,6,6,6) is {1,2,3,6,9}.
%e A370595 The a(1) = 1 through a(15) = 13 partitions (A = 10, B = 11, C = 12, D = 13):
%e A370595 1 . 21 22 . 33 322 71 441 55 533 B1 553 77 933
%e A370595 31 51 421 332 522 442 722 444 733 D1 B22
%e A370595 321 422 531 721 731 552 751 B21 B31
%e A370595 521 4321 4322 4332 931 4433 4443
%e A370595 5321 4431 4432 5441 5442
%e A370595 5322 5332 6332 5532
%e A370595 5421 5422 7322 6621
%e A370595 6321 6322 7421 7332
%e A370595 7321 7422
%e A370595 7521
%e A370595 8421
%e A370595 9321
%e A370595 54321
%t A370595 Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]==1&]],{n,0,30}]
%Y A370595 For no choices we have A370320, complement A239312.
%Y A370595 The version for prime factors (not all divisors) is A370594, ranks A370647.
%Y A370595 For multiple choices we have A370803, ranks A370811.
%Y A370595 These partitions have ranks A370810.
%Y A370595 A000005 counts divisors.
%Y A370595 A000041 counts integer partitions, strict A000009.
%Y A370595 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370595 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370595 A355741, A355744, A355745 choose prime factors of prime indices.
%Y A370595 A370592 counts partitions with choosable prime factors, ranks A368100.
%Y A370595 A370593 counts partitions without choosable prime factors, ranks A355529.
%Y A370595 A370804 counts non-condensed partitions with no ones, complement A370805.
%Y A370595 A370814 counts factorizations with choosable divisors, complement A370813.
%Y A370595 Cf. A355535, A355739, A355740, A367867, A367904, A368110, A370583, A370584, A370806, A370808.
%K A370595 nonn
%O A370595 0,5
%A A370595 _Gus Wiseman_, Mar 03 2024
%E A370595 More terms from _Jinyuan Wang_, Feb 14 2025