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A025052
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Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).
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18
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1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| According to Borwein and Choi, if the Generalized Riemann Hypothesis is true, then this sequence has no larger terms, otherwise there may be one term greater than 10^11. - T. D. Noe (noe(AT)sspectra.com), Apr 08 2004
Note that n+1 must be prime for all n in this sequence. - T. D. Noe (noe(AT)sspectra.com), Apr 28 2004
Borwein and Choi prove (Theorem 6.2) that the equation N=xy+xz+yz has an integer solution x,y,z>0 if N contains a square factor and N is not 4 or 18. In the following simple proof explicit solutions are given. Let N=mn^2, m,n integer, m>0, n>1. If n<m+1: x=n, y=n(n-1), z=m+1-n. If n=m+1, n>3: x=6, y=n-3, z=n^2-4n+6. If n>m+1: if n=0 (mod m+1): x=m+1, y=m(m+1), z=m(n^2/(m+1)^2-1), if n=k (mod m+1), 0<k<m+1 : x=k, y=m+1-k, z=m(n^2-k^2)/(m+1)+k(k-1) - Herm Jan Brascamp (brashoek(AT)hi.nl), May 28 2007
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REFERENCES
| Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158
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LINKS
| J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158 (dvi, ps).
M. Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms
J. Borwein & K.-K. S. Choi, On the Representations of xy + yz + zx
Experimental Mathematics, Home Page
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MATHEMATICA
| n=500; lim=Ceiling[(n-1)/2]; lst={}; Do[m=a*b+a*c+b*c; If[m<=n, lst=Union[lst, {m}]], {a, lim}, {b, lim}, {c, lim}]; Complement[Range[n], lst]
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CROSSREFS
| Cf. A027563, A027564, A027565, A027566, A055745, A034168.
Cf. A000926 (numbers not of the form ab+ac+bc, 0<a<b<c), A093669 (numbers having a unique representation as ab+ac+bc, 0<a<b<c), A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).
Cf. A094379, A094380, A094381.
Sequence in context: A079961 A144023 A018164 * A142584 A098197 A175941
Adjacent sequences: A025049 A025050 A025051 * A025053 A025054 A025055
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KEYWORD
| nonn,fini,nice
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Corrected by R. H. Hardin (rhhardin(AT)att.net)
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