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%I #5 Mar 16 2024 21:40:58
%S 0,0,1,1,1,3,2,4,6,7,11,11,17,20,26,34,44,56,67,84,102,131,156,195,
%T 232,283,346,411,506,598,721,855,1025,1204,1448,1689,2018,2363,2796,
%U 3265,3840,4489,5242,6104,7106,8280,9595,11143,12862,14926,17197,19862,22841
%N Number of integer partitions of n with fewer parts than distinct divisors of parts.
%C The Heinz numbers of these partitions are given by A371168.
%e The partition (4,3,2) has 3 parts {2,3,4} and 4 distinct divisors of parts {1,2,3,4}, so is counted under a(9).
%e The a(2) = 1 through a(10) = 11 partitions:
%e (2) (3) (4) (5) (6) (7) (8) (9) (10)
%e (3,2) (4,2) (4,3) (4,4) (5,4) (6,4)
%e (4,1) (5,2) (5,3) (6,3) (7,3)
%e (6,1) (6,2) (7,2) (8,2)
%e (4,3,1) (8,1) (9,1)
%e (6,1,1) (4,3,2) (4,3,3)
%e (6,2,1) (5,3,2)
%e (5,4,1)
%e (6,2,2)
%e (6,3,1)
%e (8,1,1)
%t Table[Length[Select[IntegerPartitions[n],Length[#] < Length[Union@@Divisors/@#]&]],{n,0,30}]
%Y The RHS is represented by A370820.
%Y The version for equality is A371130 (ranks A370802), strict A371128.
%Y For submultisets instead of parts on the LHS we get ranks A371166.
%Y These partitions are ranked by A371168.
%Y The opposite version is A371171, ranks A370348.
%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, A370809, A370813, A370814.
%K nonn
%O 0,6
%A _Gus Wiseman_, Mar 16 2024