A368339 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is twice a positive square. A368340 gives values of x.
1, 0, 1, 3, 3, 1, 2, 0, 2, 1, 21, 5, 5, 12, 1, 51, 3, 0, 3, 57, 1, 15, 1794, 7, 7, 45, 33, 1, 1287, 2, 4, 0, 4, 2, 15, 1, 215, 3315, 3, 9, 9, 3, 561, 4137, 1, 60, 110532, 245, 5, 0, 5, 255, 1557945, 65, 1, 6, 48, 455, 14127, 11, 11, 207480, 20, 29427, 285, 1
Offset: 1
Examples
For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).
Links
- Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.
Programs
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PARI
pellsolve(n)={if(issquare(n/2), return(0), q=bnfinit('x^2-8*n, 1); i=-1; until(y&&x==floor(x)&&y==floor(y)&&x^2-8*n*y^2==1, f=lift(q.fu[1]^i); x=abs(polcoeff(f, 0)); y=abs(polcoeff(f, 1)); i++); return(y))};