%I #7 Mar 16 2024 21:41:09

%S 0,0,1,1,2,4,5,9,12,18,26,34,50,65,92,121,161,209,274,353,456,590,745,

%T 950,1195,1507,1885,2350,2923,3611,4465,5485,6735,8223,10050,12195,

%U 14822,17909,21653,26047,31340,37557,44990,53708,64068,76241,90583,107418

%N Number of integer partitions of n with more parts than distinct divisors of parts.

%C The Heinz numbers of these partitions are given by A370348.

%e The partition (3,2,1,1) has 4 parts {1,2,3,4} and 3 distinct divisors of parts {1,2,3}, so is counted under a(7).

%e The a(0) = 0 through a(8) = 12 partitions:

%e . . (11) (111) (211) (221) (222) (331) (2222)

%e (1111) (311) (2211) (511) (3221)

%e (2111) (3111) (2221) (3311)

%e (11111) (21111) (3211) (4211)

%e (111111) (4111) (5111)

%e (22111) (22211)

%e (31111) (32111)

%e (211111) (41111)

%e (1111111) (221111)

%e (311111)

%e (2111111)

%e (11111111)

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

%Y The partitions are ranked by A370348.

%Y The opposite version is A371173, ranked by A371168.

%Y The RHS is represented by A370820, positions of twos A371127.

%Y The version for equality is A371130 (ranks A370802), strict A371128.

%Y For submultisets instead of parts on the LHS we get ranks A371167.

%Y A000005 counts divisors.

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

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

%K nonn

%O 1,5

%A _Gus Wiseman_, Mar 16 2024