A366845 Number of integer partitions of n that contain at least one even part and whose halved even parts are relatively prime.

0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 31, 43, 58, 82, 107, 144, 189, 250, 323, 420, 537, 695, 880, 1114, 1404, 1774, 2210, 2759, 3423, 4239, 5223, 6430, 7869, 9640, 11738, 14266, 17297, 20950, 25256, 30423, 36545, 43824, 52421, 62620, 74599, 88802, 105431

Offset: 0

Author

Gus Wiseman, Oct 28 2023

Keywords

• nonn

Examples

			The partition y = (6,4) has halved even parts (3,2) which are relatively prime, so y is counted under a(10).
The a(2) = 1 through a(9) = 15 partitions:
  (2)  (21)  (22)   (32)    (42)     (52)      (62)       (72)
             (211)  (221)   (222)    (322)     (332)      (432)
                    (2111)  (321)    (421)     (422)      (522)
                            (2211)   (2221)    (521)      (621)
                            (21111)  (3211)    (2222)     (3222)
                                     (22111)   (3221)     (3321)
                                     (211111)  (4211)     (4221)
                                               (22211)    (5211)
                                               (32111)    (22221)
                                               (221111)   (32211)
                                               (2111111)  (42111)
                                                          (222111)
                                                          (321111)
                                                          (2211111)
                                                          (21111111)
		

Crossrefs

For all parts we have A000837, complement A018783.

These partitions have ranks A366847.

For odd parts we have A366850, ranks A366846, complement A366842.

A000041 counts integer partitions, strict A000009, complement A047967.

A035363 counts partitions into all even parts, ranks A066207.

A078374 counts relatively prime strict partitions.

A168532 counts partitions by gcd.

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

A366531 = 2*A366533 adds up even prime indices, triangle A113686/A174713.

Cf. A051424, A113685, A116598, A258117, A289509, A366843, A366849.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], GCD@@Select[#,EvenQ]/2==1&]],{n,0,30}]