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%I #5 Mar 22 2024 09:16:47
%S 0,1,1,2,3,4,5,8,9,11,12,16,18,23,25,32,36,42,47,57,62,73,81,96,106,
%T 123,132,154,168,190,207,240,259,293,317,359,388,434,469,529,574,635,
%U 688,768,826,915,987,1093,1181,1302,1397,1540,1662,1818,1959,2149,2309
%N Number of integer partitions of n whose distinct parts form the set of divisors of some number.
%C The Heinz numbers of these partitions are given by A371288.
%e The partition y = (10,5,5,5,2,2,1) has distinct parts {1,2,5,10}, which form the set of divisors of 10, so y is counted under a(30).
%e The a(1) = 1 through a(8) = 9 partitions:
%e (1) (11) (21) (31) (221) (51) (331) (71)
%e (111) (211) (311) (2211) (421) (3311)
%e (1111) (2111) (3111) (511) (4211)
%e (11111) (21111) (2221) (5111)
%e (111111) (22111) (22211)
%e (31111) (221111)
%e (211111) (311111)
%e (1111111) (2111111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n], Union[#]==Divisors[Max[#]]&]],{n,0,30}]
%Y The strict case is A054973, ranks A371283 (unsorted version A275700).
%Y These partitions have ranks A371288.
%Y A000005 counts divisors, row-lengths of A027750.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y Cf. A001221, A002865, A239312, A370803, A371172, A371286, A371285.
%K nonn
%O 0,4
%A _Gus Wiseman_, Mar 22 2024
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