A078374 Number of partitions of n into distinct and relatively prime parts.

1, 0, 1, 1, 2, 2, 4, 4, 6, 7, 11, 10, 17, 17, 23, 26, 37, 36, 53, 53, 70, 77, 103, 103, 139, 147, 184, 199, 255, 260, 339, 358, 435, 474, 578, 611, 759, 810, 963, 1045, 1259, 1331, 1609, 1726, 2015, 2200, 2589, 2762, 3259, 3509, 4058, 4416, 5119, 5488, 6364, 6882

Offset

1,5

Comments

The Heinz numbers of these partitions are given by A302796, which is the intersection of A005117 (strict) and A289509 (relatively prime). - Gus Wiseman, Oct 18 2020

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms

Formula

Moebius transform of A000009.

G.f.: 1 + Sum_{n>=1} a(n)*x^n/(1 - x^n) = Product_{n>=1} (1 + x^n). - Ilya Gutkovskiy, Apr 26 2017

Example

From Gus Wiseman, Oct 18 2020: (Start)
The a(1) = 1 through a(13) = 17 partitions (empty column indicated by dot, A = 10, B = 11, C = 12):
  1   .  21   31   32   51    43    53    54    73     65     75     76
                   41   321   52    71    72    91     74     B1     85
                              61    431   81    532    83     543    94
                              421   521   432   541    92     651    A3
                                          531   631    A1     732    B2
                                          621   721    542    741    C1
                                                4321   632    831    643
                                                       641    921    652
                                                       731    5421   742
                                                       821    6321   751
                                                       5321          832
                                                                     841
                                                                     931
                                                                     A21
                                                                     5431
                                                                     6421
                                                                     7321
(End)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&GCD@@#==1&]], {n, 15}] (* Gus Wiseman, Oct 18 2020 *)

Crossrefs

Cf. A047966.

A000837 is the not necessarily strict version.

A302796 gives the Heinz numbers of these partitions.

A305713 is the pairwise coprime instead of relatively prime version.

A332004 is the ordered version.

A337452 is the case without 1's.

A000009 counts strict partitions.

A000740 counts relatively prime compositions.

Cf. A007359, A101268, A289508, A289509, A291166, A298748, A337451, A337485, A337451, A337561, A337563.

Sequence in context: A122134 A035940 A067772 * A366129 A341697 A242984

Adjacent sequences: A078371 A078372 A078373 * A078375 A078376 A078377

Keyword

nonn

Author

Vladeta Jovovic, Dec 24 2002

Status

approved