A062536 -id:A062536 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Colin Mallows, Dec 20 2001

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.
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).
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

			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. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A060790, A062536 (and A007875 for xy = n).

Cf. A000196, A066955.

a066360 n = length [(x,y,z) | x <- [1 .. a000196 n],
                              y <- [x .. div n x],
                              z <- [y .. n - x*y],
                              x*y+(x+y)*z == n, gcd (gcd x y) z == 1]
-- Reinhard Zumkeller, Mar 23 2012

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 *)

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

A265189 Soddy circles: the two circles tangent to each of three mutually tangent circles.

Original entry on oeis.org

69, 46, 23, 6, 138, 70, 30, 21, 5, 105, 132, 33, 11, 4, -132, 138, 92, 46, 12, 276, 140, 60, 42, 10, 210, 153, 136, 72, 17, 306, 207, 138, 69, 18, 414, 210, 90, 63, 15, 315, 216, 135, 24, 10, -135, 238, 119, 102, 21, 357, 252, 63, 28, 9, 0

Offset: 1

Colin Barker, Dec 04 2015

For any three mutually tangent circles (with radii a, b, and c), one can construct a fourth circle (the inner Soddy circle, with radius d) that is mutually tangent internally to the three circles, and a fifth circle (the outer Soddy circle, with radius e) that is mutually tangent externally to the three circles. For this sequence all five radii have integral lengths.
The sequence is an array of 5-tuples (a,b,c,d,e) ordered by increasing values of a, with a > b > c.
A positive value for the outer Soddy circle indicates that it contains the three circles; a negative value indicates that it is exterior to the three circles; a value of 0 indicates that it has an infinite radius, that is, it is a straight line.

Colin Barker, Table of n, a(n) for n = 1..1000
Kival Ngaokrajang, Illustration of a(1) - a(5), a(41) - a(45) and a(51) - a(55)
Eric Weisstein's World of Mathematics, Soddy Circles
Wikipedia, Descartes' theorem

Cf. A256694.
See also the many sequences arising from Apollonian circle packing: A135849, A137246, A154636, etc.
Also the sequences related to Soddy's circle packings: A046159, A046160, A062536, etc.

soddy(amax) = {
  my(L=List(), abc, t, u);
  for(a=1, amax,
    for(b=1, a-1,
      for(c=1, b-1,
        abc=a*b*c;
        if(issquare(abc*(a+b+c), &t),
          u=a*b+a*c+b*c;
          if(abc%(u+2*t) == 0,
            if(u-2*t != 0,
              if(abc%(u-2*t) == 0,
                listput(L, [a,b,c,abc\(u+2*t),-abc\(u-2*t)])
              )
            ,
              listput(L, [a,b,c,abc\(u+2*t),0])
            )
          )
        )
      )
    )
  );
  Vec(L)
}
soddy(253)

A256694 The radius of the largest of four circles with different integer radii arranged so that each circle is tangent externally to the other three circles.

Original entry on oeis.org

69, 70, 132, 138, 140, 153, 198, 207, 210, 216, 238, 252, 264, 264, 264, 270, 276, 280, 285, 290, 306, 345, 350, 390, 396, 396, 414, 420, 429, 432, 459, 476, 483, 490, 504, 504, 520, 528, 528, 528, 539, 540, 552, 560, 567, 570, 580, 594, 595, 612, 621, 630

Offset: 1

Colin Barker, Apr 08 2015

nonn

Kival Ngaokrajang, Illustration of the first three solutions
Eric Weisstein's World of Mathematics, Soddy Circles.

See also the many sequences arising from Apollonian circle packing: A135849, A137246, A154636, etc.

Also the sequences related to Soddy's circle packings: A046159, A046160, A062536, etc.

soddy(k) = {
  s=[];
  for(a=1, k,
    for(b=1, a-1,
      for(c=1, b-1,
        if(issquare(a*b*c*(a+b+c), &t),
          if(a*b*c % (a*b+a*c+b*c+2*t) == 0,
            s=concat(s, a)
          )
        )
      )
    )
  );
  s
}
soddy(500)

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

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A265189 Soddy circles: the two circles tangent to each of three mutually tangent circles.

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A256694 The radius of the largest of four circles with different integer radii arranged so that each circle is tangent externally to the other three circles.

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PARI