A366850 Number of integer partitions of n whose odd parts are relatively prime.

0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 32, 43, 60, 80, 110, 140, 194, 244, 327, 410, 544, 670, 883, 1081, 1401, 1708, 2195, 2651, 3382, 4069, 5129, 6157, 7708, 9194, 11438, 13599, 16788, 19911, 24432, 28858, 35229, 41507, 50359, 59201, 71489, 83776, 100731, 117784

Offset: 0

Author

Gus Wiseman, Oct 28 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(8) = 16 partitions:
  (1)  (11)  (21)   (31)    (41)     (51)      (61)       (53)
             (111)  (211)   (221)    (321)     (331)      (71)
                    (1111)  (311)    (411)     (421)      (431)
                            (2111)   (2211)    (511)      (521)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

For all parts (not just odd) we have A000837, complement A018783.

The complement is counted by A366842.

These partitions have ranks A366846.

A000041 counts integer partitions, strict A000009 (also into odds).

A000740 counts relatively prime compositions.

A078374 counts relatively prime strict partitions.

A113685 counts partitions by sum of odd parts, rank statistic A366528.

A168532 counts partitions by gcd.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

Cf. A007359, A047967, A051424, A066208, A116598, A365067, A366843, A366844, A366845, A366848.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],GCD@@Select[#,OddQ]==1&]],{n,0,30}]