A101271 - OEIS

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

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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.