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

0,4

Links

Table of n, a(n) for n=0..47.

Example

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)

Mathematica

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

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.

Sequence in context: A241725 A022480 A308927 * A024791 A178240 A118084

Adjacent sequences: A366847 A366848 A366849 * A366851 A366852 A366853

Keyword

nonn

Author

Gus Wiseman, Oct 28 2023

Status

approved