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%I #5 Mar 18 2024 09:53:26
%S 0,0,1,1,1,3,2,4,4,7,8,10,12,15,19,22,29,33,40,47,57,68,81,95,110,129,
%T 152,178,207,240,277,317,365,422,486,558,632,723,824,940,1067,1210,
%U 1371,1544,1751,1977,2233,2508,2820,3162,3555,3983,4465,4990,5571,6224
%N Number of strict integer partitions of n with fewer parts than distinct divisors of parts.
%e The strict partition (6,4,2,1) has 4 parts and 5 distinct divisors of parts {1,2,3,4,5}, so is counted under a(13).
%e The a(2) = 1 through a(11) = 10 partitions:
%e (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
%e (3,2) (4,2) (4,3) (5,3) (5,4) (6,4) (6,5)
%e (4,1) (5,2) (6,2) (6,3) (7,3) (7,4)
%e (6,1) (4,3,1) (7,2) (8,2) (8,3)
%e (8,1) (9,1) (9,2)
%e (4,3,2) (5,3,2) (10,1)
%e (6,2,1) (5,4,1) (5,4,2)
%e (6,3,1) (6,3,2)
%e (6,4,1)
%e (8,2,1)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[#]] < Length[Union@@Divisors/@#]&]],{n,0,30}]
%Y The LHS is represented by A001221, distinct case of A001222.
%Y The RHS is represented by A370820, for prime factors A303975.
%Y The version for equality is A371128.
%Y The non-strict version is A371132, ranks A371179.
%Y The non-strict complement is A371178, ranks A371177.
%Y A000005 counts divisors.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length.
%Y Cf. A003963, A239312, A319055, A355529, A370803, A370808, A370813, A371130 (A370802), A371171, A371172 (A371165), A371173 (A371168).
%K nonn
%O 0,6
%A _Gus Wiseman_, Mar 18 2024