A370320 Number of non-condensed integer partitions of n, or partitions where it is not possible to choose a different divisor of each part.

0, 0, 1, 1, 2, 4, 6, 9, 13, 20, 28, 40, 54, 74, 102, 135, 180, 235, 310, 397, 516, 658, 843, 1066, 1349, 1687, 2119, 2634, 3273, 4045, 4995, 6128, 7517, 9171, 11181, 13579, 16457, 19884, 23992, 28859, 34646, 41506, 49634, 59211, 70533, 83836, 99504, 117867

Offset: 0

Author

Gus Wiseman, Mar 02 2024

Keywords

• nonn

Comments

Includes all partitions containing 1.

Examples

			The a(0) = 0 through a(8) = 13 partitions:
  .  .  (11)  (111)  (211)   (221)    (222)     (331)      (611)
                     (1111)  (311)    (411)     (511)      (2222)
                             (2111)   (2211)    (2221)     (3221)
                             (11111)  (3111)    (3211)     (3311)
                                      (21111)   (4111)     (4211)
                                      (111111)  (22111)    (5111)
                                                (31111)    (22211)
                                                (211111)   (32111)
                                                (1111111)  (41111)
                                                           (221111)
                                                           (311111)
                                                           (2111111)
                                                           (11111111)
		

Crossrefs

The complement is counted by A239312 (condensed partitions).

These partitions have ranks A355740.

Factorizations in the case of prime factors are A368413, complement A368414.

The complement for prime factors is A370592, ranks A368100.

The version for prime factors (not all divisors) is A370593, ranks A355529.

For a unique choice we have A370595, ranks A370810.

For multiple choices we have A370803, ranks A370811.

The case without ones is A370804, complement A370805.

The version for factorizations is A370813, complement A370814.

A000005 counts divisors.

A000041 counts integer partitions.

A027746 lists prime factors, A112798 indices, length A001222.

A355731 counts choices of a divisor of each prime index, firsts A355732.

A355741 chooses prime factors of prime indices, variations A355744, A355745.

Cf. A355535, A355739, A367867, A368097, A368110, A370583, A370584, A370594, A370806, A370807, A370808.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[Select[Tuples[Divisors/@#], UnsameQ@@#&]]==0&]],{n,0,30}]

Extensions

a(31)-a(47) from Alois P. Heinz, Mar 03 2024