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A025052 Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete). 21
1, 2, 4, 6, 10, 18, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462 (list; graph; refs; listen; history; text; internal format)



Subsequence of A000926.

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


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.


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: A144023 A018164 A321403 * A142584 A098197 A175941

Adjacent sequences:  A025049 A025050 A025051 * A025053 A025054 A025055




Clark Kimberling


Corrected by R. H. Hardin



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Last modified March 29 15:16 EDT 2020. Contains 333107 sequences. (Running on oeis4.)