The OEIS is supported by the many generous donors to the OEIS Foundation.
%I #16 May 24 2024 03:49:57
%S 0,0,1,1,2,4,6,9,13,20,28,40,54,74,102,135,180,235,310,397,516,658,
%T 843,1066,1349,1687,2119,2634,3273,4045,4995,6128,7517,9171,11181,
%U 13579,16457,19884,23992,28859,34646,41506,49634,59211,70533,83836,99504,117867
%N Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.
%C Includes all partitions containing 1.
%e The a(0) = 0 through a(8) = 13 partitions:
%e . . (11) (111) (211) (221) (222) (331) (611)
%e (1111) (311) (411) (511) (2222)
%e (2111) (2211) (2221) (3221)
%e (11111) (3111) (3211) (3311)
%e (21111) (4111) (4211)
%e (111111) (22111) (5111)
%e (31111) (22211)
%e (211111) (32111)
%e (1111111) (41111)
%e (221111)
%e (311111)
%e (2111111)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]],{n,0,30}]
%Y The complement is counted by A239312 (condensed partitions).
%Y These partitions have ranks A355740.
%Y Factorizations in the case of prime factors are A368413, complement A368414.
%Y The complement for prime factors is A370592, ranks A368100.
%Y The version for prime factors (not all divisors) is A370593, ranks A355529.
%Y For a unique choice we have A370595, ranks A370810.
%Y For multiple choices we have A370803, ranks A370811.
%Y The case without ones is A370804, complement A370805.
%Y The version for factorizations is A370813, complement A370814.
%Y A000005 counts divisors.
%Y A000041 counts integer partitions.
%Y A027746 lists prime factors, A112798 indices, length A001222.
%Y A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A355741 chooses prime factors of prime indices, variations A355744, A355745.
%Y Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.
%K nonn
%O 0,5
%A _Gus Wiseman_, Mar 02 2024
%E a(31)-a(47) from _Alois P. Heinz_, Mar 03 2024