1,2

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

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

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

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]

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 this_sequence A142584 A098197 A102477

Adjacent sequences: A025049 A025050 A025051 this_sequence A025053 A025054 A025055

nonn,fini,nice

Clark Kimberling (ck6(AT)evansville.edu)

Corrected by R. H. Hardin (rhhardin(AT)att.net)