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A101271 Number of partitions of n into 3 distinct and relatively prime parts. 21
1, 1, 2, 3, 4, 5, 6, 8, 9, 12, 12, 16, 15, 21, 20, 26, 25, 33, 28, 40, 36, 45, 42, 56, 44, 65, 56, 70, 64, 84, 66, 96, 81, 100, 88, 120, 90, 133, 110, 132, 121, 161, 120, 175, 140, 176, 156, 208, 153, 220, 180, 222, 196, 261, 184, 280, 225, 270, 240, 312, 230, 341, 272 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,3

COMMENTS

The Heinz numbers of these partitions are the intersection of A289509 (relatively prime), A005117 (strict), and A014612 (triple). - Gus Wiseman, Oct 15 2020

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 6..10000

FORMULA

G.f. for the number of partitions of n into m distinct and relatively prime parts is Sum(moebius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k), i=1..m), k=1..infinity).

EXAMPLE

For n=10 we have 4 such partitions: 1+2+7, 1+3+6, 1+4+5 and 2+3+5.

From Gus Wiseman, Oct 13 2020: (Start)

The a(6) = 1 through a(18) = 15 triples (A..F = 10..15):

  321  421  431  432  532  542  543  643  653  654  754  764  765

            521  531  541  632  651  652  743  753  763  854  873

                 621  631  641  732  742  752  762  853  863  954

                      721  731  741  751  761  843  871  872  972

                           821  831  832  851  852  943  953  981

                                921  841  932  861  952  962  A53

                                     931  941  942  961  971  A71

                                     A21  A31  951  A51  A43  B43

                                          B21  A32  B32  A52  B52

                                               A41  B41  A61  B61

                                               B31  C31  B42  C51

                                               C21  D21  B51  D32

                                                         C32  D41

                                                         C41  E31

                                                         D31  F21

                                                         E21

(End)

MAPLE

m:=3: with(numtheory): g:=sum(mobius(k)*x^(m*(m+1)/2*k)/Product(1-x^(i*k), i=1..m), k=1..20): gser:=series(g, x=0, 80): seq(coeff(gser, x^n), n=6..77); # Emeric Deutsch, May 31 2005

MATHEMATICA

Table[Length[Select[IntegerPartitions[n, {3}], UnsameQ@@#&&GCD@@#==1&]], {n, 6, 50}] (* Gus Wiseman, Oct 13 2020 *)

CROSSREFS

Cf. A023024-A023030, A000742-A000743, A023031-A023035.

A000741 is the ordered non-strict version.

A001399(n-6) does not require relative primality.

A023022 counts pairs instead of triples.

A023023 is the not necessarily strict version.

A078374 counts these partitions of any length, with Heinz numbers A302796.

A101271*6 is the ordered version.

A220377 is the pairwise coprime instead of relatively prime version.

A284825 counts the case that is pairwise non-coprime also.

A337605 is the pairwise non-coprime instead of relatively prime version.

A008289 counts strict partitions by sum and length.

A007304 gives the Heinz numbers of 3-part strict partitions.

A307719 counts 3-part pairwise coprime partitions.

A337601 counts 3-part partitions whose distinct parts are pairwise coprime.

Cf. A000010, A000217, A000837, A007360, A014612, A055684, A289509, A332004, A337452, A337563.

Sequence in context: A235592 A100054 A330193 * A093110 A165707 A052063

Adjacent sequences:  A101268 A101269 A101270 * A101272 A101273 A101274

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Dec 19 2004

EXTENSIONS

More terms from Emeric Deutsch, May 31 2005

STATUS

approved

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)