A067873 - OEIS

%I #13 Jun 15 2018 02:55:24

%S 7,9,15,17,19,23,25,26,31,33,35,37,39,41,47,49,51,53,55,57,63,65,71,

%T 73,79,80,81,82,87,89,91,95,97,99,101,103,105,107,109,111,113,119,121,

%U 125,127,129,135,137,143,145,149,151,153,159,161,163,167,169,170,175,177

%N Numbers x satisfying x^2 - D*y^2 = 1 for more than one value of D distinct from x^2 - 1.

%C The sequence contains A056220(n) for n>1. For a given x, solutions (x,y) to x^2 - D*y^2 = 1 do not all stand for the least nontrivial positive solution pair. For instance, x=244 solves the latter equation for D=15 and D=135 with corresponding y=63 and y=21. While (244,21) is indeed the fundamental solution to x^2 - 135*y^2 = 1, (244,63) comes only as the third one after the solution pairs (4,1) and (31,8) to x^2 - 15*y^2 = 1.

%C Sequence includes mostly odd entries in consecutive pairs. It seems reasonable to conjecture that an even entry e satisfies big omega(E) - omega(E) > 2, where E = e^2 - 1. (See A001222 and A001221.)

%o (PARI) { for(x=2,10^3,f=core(x^2-1,1)[2];if(f>1 && !isprime(f),print1(x,", ")) ) } \\ _Max Alekseyev_, Jul 25 2009

%Y Cf. A056220.

%K nonn

%O 1,1

%A _Lekraj Beedassy_, Feb 25 2002

%E Corrected and extended by _Max Alekseyev_, Jul 25 2009