cp's OEIS Frontend

A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.

%I A371130 #5 Mar 17 2024 15:10:36
%S A371130 1,1,0,1,2,0,4,2,4,5,5,11,10,16,17,21,26,32,44,53,69,71,101,110,148,
%T A371130 168,205,249,289,356,418,502,589,716,812,999,1137,1365,1566,1873,2158,
%U A371130 2537,2942,3449,4001,4613,5380,6193,7220,8224,9575,10926,12683,14430
%N A371130 Number of integer partitions of n such that the number of parts is equal to the number of distinct divisors of parts.
%C A371130 The Heinz numbers of these partitions are given by A370802.
%e A371130 The partition (6,2,2,1) has 4 parts and 4 distinct divisors of parts {1,2,3,6} so is counted under a(11).
%e A371130 The a(1) = 1 through a(11) = 11 partitions:
%e A371130   (1)  .  (21)  (22)  .  (33)   (322)  (71)   (441)   (55)    (533)
%e A371130                 (31)     (51)   (421)  (332)  (522)   (442)   (722)
%e A371130                          (321)         (422)  (531)   (721)   (731)
%e A371130                          (411)         (521)  (4311)  (4321)  (911)
%e A371130                                               (6111)  (6211)  (4322)
%e A371130                                                               (4331)
%e A371130                                                               (5321)
%e A371130                                                               (5411)
%e A371130                                                               (6221)
%e A371130                                                               (6311)
%e A371130                                                               (8111)
%t A371130 Table[Length[Select[IntegerPartitions[n], Length[#]==Length[Union@@Divisors/@#]&]],{n,0,30}]
%Y A371130 The LHS is represented by A001222, distinct A000021.
%Y A371130 These partitions are ranked by A370802.
%Y A371130 The RHS is represented by A370820, for prime factors A303975.
%Y A371130 The strict case is A371128.
%Y A371130 For (greater than) instead of (equal to) we have A371171, ranks A370348.
%Y A371130 For submultisets instead of parts on the LHS we have A371172.
%Y A371130 For (less than) instead of (equal to) we have A371173, ranked by A371168.
%Y A371130 Counting only distinct parts on the LHS gives A371178, ranks A371177.
%Y A371130 A000005 counts divisors.
%Y A371130 A000041 counts integer partitions, strict A000009.
%Y A371130 A008284 counts partitions by length.
%Y A371130 Choosable partitions: A239312 (A368110), A355740 (A370320), A370592 (A368100), A370593 (A355529).
%Y A371130 Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.
%K A371130 nonn
%O A371130 0,5
%A A371130 _Gus Wiseman_, Mar 17 2024