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%I A370320 #16 May 24 2024 03:49:57
%S A370320 0,0,1,1,2,4,6,9,13,20,28,40,54,74,102,135,180,235,310,397,516,658,
%T A370320 843,1066,1349,1687,2119,2634,3273,4045,4995,6128,7517,9171,11181,
%U A370320 13579,16457,19884,23992,28859,34646,41506,49634,59211,70533,83836,99504,117867
%N A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.
%C A370320 Includes all partitions containing 1.
%e A370320 The a(0) = 0 through a(8) = 13 partitions:
%e A370320   .  .  (11)  (111)  (211)   (221)    (222)     (331)      (611)
%e A370320                      (1111)  (311)    (411)     (511)      (2222)
%e A370320                              (2111)   (2211)    (2221)     (3221)
%e A370320                              (11111)  (3111)    (3211)     (3311)
%e A370320                                       (21111)   (4111)     (4211)
%e A370320                                       (111111)  (22111)    (5111)
%e A370320                                                 (31111)    (22211)
%e A370320                                                 (211111)   (32111)
%e A370320                                                 (1111111)  (41111)
%e A370320                                                            (221111)
%e A370320                                                            (311111)
%e A370320                                                            (2111111)
%e A370320                                                            (11111111)
%t A370320 Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]],{n,0,30}]
%Y A370320 The complement is counted by A239312 (condensed partitions).
%Y A370320 These partitions have ranks A355740.
%Y A370320 Factorizations in the case of prime factors are A368413, complement A368414.
%Y A370320 The complement for prime factors is A370592, ranks A368100.
%Y A370320 The version for prime factors (not all divisors) is A370593, ranks A355529.
%Y A370320 For a unique choice we have A370595, ranks A370810.
%Y A370320 For multiple choices we have A370803, ranks A370811.
%Y A370320 The case without ones is A370804, complement A370805.
%Y A370320 The version for factorizations is A370813, complement A370814.
%Y A370320 A000005 counts divisors.
%Y A370320 A000041 counts integer partitions.
%Y A370320 A027746 lists prime factors, A112798 indices, length A001222.
%Y A370320 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370320 A355741 chooses prime factors of prime indices, variations A355744, A355745.
%Y A370320 Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.
%K A370320 nonn
%O A370320 0,5
%A A370320 _Gus Wiseman_, Mar 02 2024
%E A370320 a(31)-a(47) from _Alois P. Heinz_, Mar 03 2024

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