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 A371172 #9 Mar 16 2024 21:41:04
%S A371172 0,0,1,1,0,1,0,3,2,3,1,4,2,1,2,3,4,2,4,1,5,2,7,5,9,4,9,15,18,16,24,13,
%T A371172 17,23,23,22,34,17,30,31,36,29,43,21,30,35,44,28,47,19,44
%N A371172 Number of integer partitions of n with as many submultisets as distinct divisors of parts.
%C A371172 The Heinz numbers of these partitions are given by A371165.
%e A371172 The partition (8,6,6) has 6 submultisets {(8,6,6),(8,6),(6,6),(8),(6),()} and 6 distinct divisors of parts {1,2,3,4,6,8}, so is counted under a(20).
%e A371172 The a(17) = 2 through a(24) = 9 partitions:
%e A371172 (17) (9,9) (19) (11,9) (14,7) (13,9) (23) (21,3)
%e A371172 (13,4) (15,3) (15,5) (17,4) (21,1) (19,4) (22,2)
%e A371172 (6,6,6) (8,6,6) (8,8,6) (22,1) (8,8,8)
%e A371172 (12,3,3) (12,4,4) (10,6,6) (15,4,4) (10,8,6)
%e A371172 (18,1,1) (16,3,3) (12,10,1) (12,6,6)
%e A371172 (18,2,2) (12,7,5)
%e A371172 (20,1,1) (18,3,3)
%e A371172 (20,2,2)
%e A371172 (12,10,2)
%t A371172 Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]],{n,0,30}]
%Y A371172 The RHS is represented by A370820.
%Y A371172 Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.
%Y A371172 These partitions are ranked by A371165.
%Y A371172 A000005 counts divisors.
%Y A371172 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A371172 Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
%Y A371172 Cf. A003963, A319055, A355739, A370803, A370808, A370813, A370814, A371166.
%K A371172 nonn
%O A371172 0,8
%A A371172 _Gus Wiseman_, Mar 16 2024