%I A094185
%S A094185 1,6,12,24,24,49,41,59,61,100,56,132,76,127,122,165,87,223,95,207,176,
%T A094185 190,110,318,187,232,211,342,137,440,127,327
%N A094185 Number of solutions to n == xy (mod z) == yz (mod x) == zx (mod y) with 
               0 < x < y < z.
%C A094185 Possner (see also Knuth) asks for solutions when n=2. There is at least 
               one solution for all positive n: (x,y,z) = (n+1, n^2+2n, n^3+3n^2+n). 
               All solutions appear to be in the polytope n < x <= 2n^2+n, x < y 
               <= 2n^3+2n^2-n, y < z <= n^5+2n^4+2n^3+n^2-n. Many solutions, especially 
               for prime n, are such that n divides x, y and z. See A094595.
%D A094185 M. F. Possner, Problem 11021, Amer. Math. Monthly, 110 (2003), p. 542
%D A094185 D. E. Knuth, Solution 11021, Amer. Math. Monthly, 112 (2005), p. 279
%e A094185 a(2) = 6 because there are 6 solutions: (x,y,z) = (3, 8, 22), (3, 10, 
               14), (4, 5, 18), (4, 6, 11), (6, 14, 82) and (6, 22, 26).
%t A094185 Table[cnt=0; Do[d=Divisors[x*y-n]; Do[z=d[[i]]; If[z>y, If[Mod[x*z, y]==n 
               && Mod[y*z, x]==n, cnt++;]], {i, Length[d]}], {x, n+1, n(2n+1)}, 
               {y, x+1, n(2n^2+2n-1)}]; cnt, {n, 10}]
%Y A094185 Cf. A094595 (number of solutions to 1 = nxy (mod z) = nyz (mod x) = nzx 
               (mod y) with 0<x<y<z).
%Y A094185 Sequence in context: A005694 A081512 A096387 this_sequence A074902 A096366 
               A061822
%Y A094185 Adjacent sequences: A094182 A094183 A094184 this_sequence A094186 A094187 
               A094188
%K A094185 more,nonn
%O A094185 1,2
%A A094185 T. D. Noe (noe(AT)sspectra.com), May 06 200, revised May 13 2004