A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).

1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset

1,2

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, Apr 08 2004

Note that n+1 must be prime for all n in this sequence. - T. D. Noe, 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

Links

Table of n, a(n) for n=1..18.

J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.

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.

M. Peters, The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.

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]

Crossrefs

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0<a<b<c) and of A006093.

Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.

Cf. 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).

Sequence in context: A018164 A321403 A340311 * A142584 A098197 A358443

Adjacent sequences: A025049 A025050 A025051 * A025053 A025054 A025055

Keyword

nonn,fini,nice

Author

Clark Kimberling

Extensions

Corrected by R. H. Hardin

Status

approved