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A066360
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Number of unordered solutions in positive integers of xy+xz+yz = n with GCD(x,y,z) = 1.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,11
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COMMENTS
| 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 (noe(AT)sspectra.com), Jun 14 2006
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LINKS
| T. D. Noe, Table of n, a(n) for n = 1..10000
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EXAMPLE
| 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).
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MATHEMATICA
| 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 (noe(AT)sspectra.com), Jun 14 2006
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CROSSREFS
| Cf. A060790, A062536 (and A007875 for xy = n).
Sequence in context: A060184 A055639 A156542 * A061358 A025866 A048881
Adjacent sequences: A066357 A066358 A066359 * A066361 A066362 A066363
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KEYWORD
| nonn,nice
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AUTHOR
| Colin Mallows (colinm(AT)avaya.com), Dec 20 2001
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EXTENSIONS
| Corrected and extended by T. D. Noe (noe(AT)sspectra.com), Jun 14 2006
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