A007360 Number of partitions of n into distinct and pairwise relatively prime parts. (Formerly M0264)

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 9, 10, 11, 10, 13, 17, 19, 21, 22, 21, 24, 32, 37, 37, 38, 40, 45, 55, 65, 69, 66, 64, 75, 86, 100, 113, 107, 106, 122, 145, 165, 174, 167, 162, 179, 222, 253, 255, 255, 255, 273, 328, 373, 376, 369, 377, 406, 476, 553, 569, 537, 529

Offset

1,3

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..750 (terms 1..350 from Alois P. Heinz)

M. LeBrun & D. Hoey, Emails

Formula

a(n) = A051424(n)-A051424(n-2). - Vladeta Jovovic, Dec 11 2004

Example

From Gus Wiseman, Sep 23 2019: (Start)
The a(1) = 1 through a(10) = 6 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)   (8)    (9)    (A)
            (21)  (31)  (32)  (51)   (43)  (53)   (54)   (73)
                        (41)  (321)  (52)  (71)   (72)   (91)
                                     (61)  (431)  (81)   (532)
                                           (521)  (531)  (541)
                                                         (721)
(End)

Mathematica

$RecursionLimit = 1000; b[n_, i_, s_] := b[n, i, s] = Module[{f}, If[n == 0 || i == 1, 1, If[i<2, 0, f = FactorInteger[i][[All, 1]]; b[n, i-1, Select[s, #<i&]] + If[i <= n && f ~Intersection~ s == {}, b[n-i, i-1, Select[s ~Union~ f, #<i&]], 0]]]]; a[n_] := b[n, n, {}] - b[n-2, n-2, {}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 20 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Length[#]==1||UnsameQ@@#&&CoprimeQ@@Union[#]&]], {n, 0, 30}] (* Gus Wiseman, Sep 23 2019 *)

Crossrefs

Number of partitions of n into relatively prime parts = A000837.

The non-strict case is A051424.

Strict relatively prime partitions are A078374.

Cf. A007359, A038348, A084422, A186974, A187106, A303140, A302569, A303362, A304714, A320426, A320436.

Sequence in context: A325393 A320388 A264396 * A359320 A029144 A173244

Adjacent sequences: A007357 A007358 A007359 * A007361 A007362 A007363

Keyword

nonn,easy

Author

N. J. A. Sloane and Mira Bernstein, following a suggestion from Marc LeBrun.

Extensions

More precise definition from Vladeta Jovovic, Dec 11 2004

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005

Status

approved