A101271 Number of partitions of n into 3 distinct and relatively prime parts.

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

Offset: 6

Author

Vladeta Jovovic, Dec 19 2004

Keywords

• easy
• nonn

Comments

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

Examples

			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)
		

Links

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

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.

Programs

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 *)

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).

Extensions

More terms from Emeric Deutsch, May 31 2005