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

Author

Clark Kimberling

Keywords

• nonn
• fini
• nice

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 n3: 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

Links

• 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.

Crossrefs

Subsequence of A000926 (numbers not of the form ab+ac+bc, 0A006093.

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

Cf. A093669 (numbers having a unique representation as ab+ac+bc, 0A093670 (numbers having a unique representation as ab+ac+bc, 0<=a<=b<=c).

Programs

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]

Extensions

Corrected by R. H. Hardin

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