A371172 - OEIS

%I #9 Mar 16 2024 21:41:04

%S 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 17,23,23,22,34,17,30,31,36,29,43,21,30,35,44,28,47,19,44

%N Number of integer partitions of n with as many submultisets as distinct divisors of parts.

%C The Heinz numbers of these partitions are given by A371165.

%e 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 The a(17) = 2 through a(24) = 9 partitions:

%e   (17)    (9,9)     (19)  (11,9)    (14,7)  (13,9)    (23)       (21,3)

%e   (13,4)  (15,3)          (15,5)    (17,4)  (21,1)    (19,4)     (22,2)

%e           (6,6,6)         (8,6,6)           (8,8,6)   (22,1)     (8,8,8)

%e           (12,3,3)        (12,4,4)          (10,6,6)  (15,4,4)   (10,8,6)

%e                           (18,1,1)          (16,3,3)  (12,10,1)  (12,6,6)

%e                                             (18,2,2)             (12,7,5)

%e                                             (20,1,1)             (18,3,3)

%e                                                                  (20,2,2)

%e                                                                  (12,10,2)

%t Table[Length[Select[IntegerPartitions[n], Length[Divisors[Times@@Prime/@#]] == Length[Union@@Divisors/@#]&]],{n,0,30}]

%Y The RHS is represented by A370820.

%Y Counting parts on the LHS gives A371130 (ranks A370802), strict A371128.

%Y These partitions are ranked by A371165.

%Y A000005 counts divisors.

%Y A355731 counts choices of a divisor of each prime index, firsts A355732.

%Y Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).

%Y Cf. A003963, A319055, A355739, A370803, A370808, A370813, A370814, A371166.

%K nonn

%O 0,8

%A _Gus Wiseman_, Mar 16 2024