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 A370803 #12 Feb 14 2025 09:45:56
%S A370803 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,
%T A370803 213,252,301,352,426,497,589,684,802,939,1095,1270,1480,1718,1985,
%U A370803 2289,2645,3056,3489,4019,4590,5289,6014,6877,7817,8955,10134,11551,13085
%N A370803 Number of integer partitions of n such that more than one set can be obtained by choosing a different divisor of each part.
%F A370803 a(n) = A239312(n) - A370595(n). - _Jinyuan Wang_, Feb 14 2025
%e A370803 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).
%e A370803 The a(0) = 0 through a(13) = 18 partitions (A..D = 10..13):
%e A370803 . . 2 3 4 5 6 7 8 9 A B C D
%e A370803 32 42 43 44 54 64 65 66 76
%e A370803 41 52 53 63 73 74 75 85
%e A370803 61 62 72 82 83 84 94
%e A370803 431 81 91 92 93 A3
%e A370803 432 433 A1 A2 B2
%e A370803 621 532 443 543 C1
%e A370803 541 542 633 544
%e A370803 622 632 642 643
%e A370803 631 641 651 652
%e A370803 821 732 661
%e A370803 741 742
%e A370803 822 832
%e A370803 831 841
%e A370803 921 922
%e A370803 A21
%e A370803 5431
%e A370803 6421
%t A370803 Table[Length[Select[IntegerPartitions[n],Length[Union[Sort /@ Select[Tuples[Divisors/@#],UnsameQ@@#&]]]>1&]],{n,0,30}]
%Y A370803 Including partitions with one choice gives A239312, complement A370320.
%Y A370803 For a unique choice we have A370595, ranks A370810.
%Y A370803 These partitions have ranks A370811.
%Y A370803 A000005 counts divisors.
%Y A370803 A000041 counts integer partitions, strict A000009.
%Y A370803 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370803 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370803 A355733 counts divisor-choices of prime indices.
%Y A370803 A355741, A355744, A355745 choose prime factors of prime indices.
%Y A370803 A370592 counts factor-choosable partitions, ranks A368100.
%Y A370803 A370593 counts non-factor-choosable partitions, ranks A355529.
%Y A370803 Cf. A355739, A368110, A370594, A370804, A370805, A370808, A370814.
%K A370803 nonn
%O A370803 0,6
%A A370803 _Gus Wiseman_, Mar 03 2024
%E A370803 More terms from _Jinyuan Wang_, Feb 14 2025