%I A067069
%S A067069 2,3,4,6,8,10,12,16,18,22,24,28,30,40,42,48,58,60,70,72,78,88,102,112,
%T A067069 120,130,168,190,210,232,240,280,312,330,408,462,520,760,840,1320,1848
%N A067069 Numbers n such that for every k the following condition holds: (*) The 
               number of nonnegative solutions of the Diophantine equation x^2 + 
               n*y^2 = 1+n*k^2 equals one half of the number of divisors of 1+n*k^2 
               if 1+n*k^2 is not a square and one half of 1 + the number of divisors 
               of 1+n*k^2 if 1+n*k^2 is a square.
%C A067069 The sequence is conjectural. It depends on a proof of conjecture 1: If 
               condition (*) holds for k and n < u(k), then it holds for k+1 and 
               n, where u(k) is some nondecreasing function of k. Computationally 
               this is confirmed for 1 < k < 300 and n < 100, 2 < k < 250 and n 
               < 238, 3 < k < 250 and n < 282, 5 < k < 200 and n < 652, 8 < k < 
               200 and n < 2088, 9 < k < 200 and n <= 1000000.
%C A067069 No further terms up to 1000000 were found; this suggests conjecture 2: 
               The sequence is finite and 1848 its last term. Perhaps easier to 
               prove is the stronger conjecture 3: For every n > c there is a k 
               < d such that condition (*) is violated for k and n. Computationally 
               this is confirmed for n <=1000000 and c = 1848, d = 11.
%o A067069 (PARI) {h=50; m=5000; v=vector(m,x,1); for(k=1,h, for(n=1,m,if(v[n]>0,
               r=0; for(x=0,sqrtint(n*k^2+1), for(y=0,sqrtint((n*k^2+1-x^2)\n),if(x^2+n*y^2==n*k^2+1,
               r++))); q=n*k^2+1; d=numdiv(q); s=if(issquare(q),d+1,d)/2; if(r!=s,
               v[n]=0)))); for(n=1,m,if(v[n]>0,print1(n,",")))}
%Y A067069 Sequence in context: A139542 A093717 A002093 this_sequence A100497 A088881 
               A020697
%Y A067069 Adjacent sequences: A067066 A067067 A067068 this_sequence A067070 A067071 
               A067072
%K A067069 nonn
%O A067069 1,1
%A A067069 Holger Stephan (stephan(AT)wias-berlin.de), Feb 18 2002
%E A067069 Edited and extended by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), 
               Jun 25 2003