

A067754


Number of unordered primitive solutions to xy+xz+yz=n.


5



1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 4, 2, 3, 2, 3, 3, 4, 2, 4, 4, 2, 4, 3, 2, 4, 4, 4, 3, 4, 3, 6, 3, 2, 4, 6, 4, 5, 4, 3, 4, 4, 3, 6, 4, 3, 4, 6, 3, 4, 4, 6, 6, 4, 2, 7, 4, 4, 5, 6, 3, 6, 6, 3, 5, 6, 4, 8, 4, 3, 6, 6, 4, 6, 4, 6, 6, 4, 3, 7, 6, 4, 6, 8, 4, 7, 6, 6, 4, 4, 5, 10, 6, 3, 5, 6, 3
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OFFSET

1,3


COMMENTS

For n = m^2 this is the number of root Descartes quadruples (m,b,c,d).
An upper bound on the number of solutions appears to be 1.5*sqrt(n).  T. D. Noe, Jun 14 2006


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
R. L. Graham, J. C. Lagarias, C. L. Mallows, Allan Wilks, and C. H. Yan, Apollonian Circle Packings: Number Theory, J. Number Theory, 100 (2003), 145.


FORMULA

a(n) = A066360(n) + A007875(n).  T. D. Noe, Jun 14 2006


EXAMPLE

a(9)=2 because of (0,1,9) and (1,1,4) (but not (0,3,3)).


MATHEMATICA

Table[cnt=0; Do[z=(nx*y)/(x+y); If[IntegerQ[z] && GCD[x, y, z]==1, cnt++ ], {x, 0, Sqrt[n/3]}, {y, Max[1, x], Sqrt[x^2+n]x}]; cnt, {n, 100}] (* T. D. Noe, Jun 14 2006 *)


CROSSREFS

Cf. A067751, A067752, A067753.
Sequence in context: A155124 A138033 A283876 * A194824 A339931 A339221
Adjacent sequences: A067751 A067752 A067753 * A067755 A067756 A067757


KEYWORD

easy,nonn


AUTHOR

Colin Mallows, Jan 31 2002


EXTENSIONS

Corrected and extended by T. D. Noe, Jun 14 2006


STATUS

approved



