%I #5 Mar 17 2024 15:10:36

%S 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 168,205,249,289,356,418,502,589,716,812,999,1137,1365,1566,1873,2158,

%U 2537,2942,3449,4001,4613,5380,6193,7220,8224,9575,10926,12683,14430

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

%C The Heinz numbers of these partitions are given by A370802.

%e 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 The a(1) = 1 through a(11) = 11 partitions:

%e (1) . (21) (22) . (33) (322) (71) (441) (55) (533)

%e (31) (51) (421) (332) (522) (442) (722)

%e (321) (422) (531) (721) (731)

%e (411) (521) (4311) (4321) (911)

%e (6111) (6211) (4322)

%e (4331)

%e (5321)

%e (5411)

%e (6221)

%e (6311)

%e (8111)

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

%Y The LHS is represented by A001222, distinct A000021.

%Y These partitions are ranked by A370802.

%Y The RHS is represented by A370820, for prime factors A303975.

%Y The strict case is A371128.

%Y For (greater than) instead of (equal to) we have A371171, ranks A370348.

%Y For submultisets instead of parts on the LHS we have A371172.

%Y For (less than) instead of (equal to) we have A371173, ranked by A371168.

%Y Counting only distinct parts on the LHS gives A371178, ranks A371177.

%Y A000005 counts divisors.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length.

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

%Y Cf. A003963, A319055, A355731, A370803, A370808, A370809, A370813, A370814.

%K nonn

%O 0,5

%A _Gus Wiseman_, Mar 17 2024