|
%I #30 Dec 19 2022 15:51:20
%S 1,2,4,6,10,18,22,30,42,58,70,78,102,130,190,210,330,462
%N Numbers not of form ab + bc + ca for 1<=a<=b<=c (probably the list is complete).
%C 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
%C Note that n+1 must be prime for all n in this sequence. - _T. D. Noe_, Apr 28 2004
%C 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
%H J. Borwein and K.-K. S. Choi, <a href="http://projecteuclid.org/euclid.em/1046889597">On the representations of xy+yz+zx</a>, Experimental Mathematics, 9 (2000), 153-158.
%H Maohua Le, <a href="http://www.math.klte.hu/publi/load_jpg.php?p=418">A note on positive integer solutions of the equation xy+yz+zx=n</a>, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.
%H M. Peters, <a href="http://projecteuclid.org/euclid.em/1103749835">The Diophantine Equation xy + yz + zx = n and Indecomposable Binary Quadratic Forms</a>, Experiment. Math., Volume 13, Issue 3 (2004), 273-274.
%t 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]
%Y Subsequence of A000926 (numbers not of the form ab+ac+bc, 0<a<b<c) and of A006093.
%Y Cf. A027563, A027564, A027565, A027566, A055745, A034168, A094379, A094380, A094381.
%Y 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).
%K nonn,fini,nice
%O 1,2
%A _Clark Kimberling_
%E Corrected by _R. H. Hardin_
|
