cp's OEIS Frontend

A066360 Number of unordered solutions in positive integers of xy + xz + yz = n with gcd(x,y,z) = 1.

%I A066360 #22 Oct 19 2019 04:10:52
%S A066360 0,0,1,0,1,0,1,1,1,0,2,0,1,1,2,1,2,0,2,1,2,0,3,2,1,2,2,0,3,0,3,2,2,1,
%T A066360 4,1,1,2,4,2,4,0,2,2,2,1,5,2,2,2,4,1,3,2,4,4,2,0,6,0,3,3,4,2,4,2,2,3,
%U A066360 4,0,7,2,2,4,4,2,4,0,5,4,3,1,6,2,2,4,6,2,6,2,4,2,2,3,8,4,2,3,4,1
%N A066360 Number of unordered solutions in positive integers of xy + xz + yz = n with gcd(x,y,z) = 1.
%C A066360 These correspond to Descartes quadruples (-s, s+x+y, s+x+z, s+y+z) where s = sqrt(n), which are primitive if n is a perfect square.
%C A066360 Many empirical regularities are known, e.g., for n = 2^(2k) or n=2^(2k-1), (2 <= k <= 10 and even k <= 20), a(n) = 2^(k-2).
%C A066360 It appears that a(n) > 0 for n > 462. An upper bound on the number of solutions appears to be 1.5*sqrt(n). - _T. D. Noe_, Jun 14 2006
%H A066360 T. D. Noe, <a href="/A066360/b066360.txt">Table of n, a(n) for n = 1..10000</a>
%e A066360 a(81) = 3 because we have the triples (x,y,z) = (1,1,40),(2,3,15),(3,6,7) (and not (3,3,12) because this is not primitive).
%t A066360 Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z] && GCD[x,y,z]==1, cnt++ ], {x,Sqrt[n/3]}, {y,x,Sqrt[x^2+n]-x}]; cnt, {n,100}] (* _T. D. Noe_, Jun 14 2006 *)
%o A066360 (Haskell)
%o A066360 a066360 n = length [(x,y,z) | x <- [1 .. a000196 n],
%o A066360                               y <- [x .. div n x],
%o A066360                               z <- [y .. n - x*y],
%o A066360                               x*y+(x+y)*z == n, gcd (gcd x y) z == 1]
%o A066360 -- _Reinhard Zumkeller_, Mar 23 2012
%Y A066360 Cf. A060790, A062536 (and A007875 for xy = n).
%Y A066360 Cf. A000196, A066955.
%K A066360 nonn,nice
%O A066360 1,11
%A A066360 _Colin Mallows_, Dec 20 2001
%E A066360 Corrected and extended by _T. D. Noe_, Jun 14 2006