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A220377 Number of partitions of n into three distinct and mutually relatively prime parts.

%I A220377 #31 Jan 12 2021 21:36:59
%S A220377 1,0,2,1,3,1,6,1,7,3,7,3,14,3,15,6,14,6,25,6,22,10,25,9,42,8,34,15,37,
%T A220377 15,53,13,48,22,53,17,78,17,65,30,63,24,99,24,88,35,84,30,126,34,103,
%U A220377 45,103,38,166,35,124,57,128,51,184,44,150,67,172,52,218
%N A220377 Number of partitions of n into three distinct and mutually relatively prime parts.
%C A220377 The Heinz numbers of these partitions are the intersection of A005117 (strict), A014612 (triples), and A302696 (coprime). - _Gus Wiseman_, Oct 14 2020
%H A220377 Fausto A. C. Cariboni, <a href="/A220377/b220377.txt">Table of n, a(n) for n = 6..10000</a> (terms 6..1000 from Seiichi Manyama)
%F A220377 a(n > 2) = A307719(n) - 1. - _Gus Wiseman_, Oct 15 2020
%e A220377 For n=10 we have three such partitions: 1+2+7, 1+4+5 and 2+3+5.
%e A220377 From _Gus Wiseman_, Oct 14 2020: (Start)
%e A220377 The a(6) = 1 through a(20) = 15 triples (empty column indicated by dot, A..H = 10..17):
%e A220377 321  .  431  531  532  731  543  751  743  753  754  971  765  B53  875
%e A220377         521       541       651       752  951  853  B51  873  B71  974
%e A220377                   721       732       761  B31  871  D31  954  D51  A73
%e A220377                             741       851       952       972       A91
%e A220377                             831       941       B32       981       B54
%e A220377                             921       A31       B41       A71       B72
%e A220377                                       B21       D21       B43       B81
%e A220377                                                           B52       C71
%e A220377                                                           B61       D43
%e A220377                                                           C51       D52
%e A220377                                                           D32       D61
%e A220377                                                           D41       E51
%e A220377                                                           E31       F41
%e A220377                                                           F21       G31
%e A220377                                                                     H21
%e A220377 (End)
%t A220377 Table[Length@Select[ IntegerPartitions[ n, {3}], #[[1]] != #[[2]] != #[[3]] && GCD[#[[1]], #[[2]]] == 1 && GCD[#[[1]], #[[3]]] == 1 && GCD[#[[2]], #[[3]]] == 1 &], {n, 6, 100}]
%t A220377 Table[Count[IntegerPartitions[n,{3}],_?(CoprimeQ@@#&&Length[ Union[#]] == 3&)],{n,6,100}] (* _Harvey P. Dale_, May 22 2020 *)
%o A220377 (PARI) a(n)=my(P=partitions(n));sum(i=1,#P,#P[i]==3&&P[i][1]<P[i][2]&&P[i][2]<P[i][3]&&gcd(P[i][1],P[i][2])==1&&gcd(P[i][1],P[i][3])==1&&gcd(P[i][2],P[i][3])==1) \\ _Charles R Greathouse IV_, Dec 14 2012
%Y A220377 Cf. A015617, A300815.
%Y A220377 A023022 is the 2-part version.
%Y A220377 A101271 is the relative prime instead of pairwise coprime version.
%Y A220377 A220377*6 is the ordered version.
%Y A220377 A305713 counts these partitions of any length, with Heinz numbers A302797.
%Y A220377 A307719 is the non-strict version.
%Y A220377 A337461 is the non-strict ordered version.
%Y A220377 A337563 is the case with no 1's.
%Y A220377 A337605 is the pairwise non-coprime instead of pairwise coprime version.
%Y A220377 A001399(n-6) counts strict 3-part partitions, with Heinz numbers A007304.
%Y A220377 A008284 counts partitions by sum and length, with strict case A008289.
%Y A220377 A318717 counts pairwise non-coprime strict partitions.
%Y A220377 A326675 ranks pairwise coprime sets.
%Y A220377 A327516 counts pairwise coprime partitions.
%Y A220377 A337601 counts 3-part partitions whose distinct parts are pairwise coprime.
%Y A220377 Cf. A000217, A007360, A023023, A051424, A078374, A087087, A302696, A333227, A337485, A337561.
%K A220377 nonn
%O A220377 6,3
%A A220377 _Carl Najafi_, Dec 13 2012