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

0,5

Links

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

Example

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)

Mathematica

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

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.

Sequence in context: A308990 A321142 A336106 * A024792 A280661 A340659

Adjacent sequences: A366842 A366843 A366844 * A366846 A366847 A366848

Keyword

nonn

Author

Gus Wiseman, Oct 28 2023

Status

approved