A368340 Take the solution to Pellian equation x^2 - 8*n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is twice a positive square. A368339 gives values of y.
3, 1, 5, 17, 19, 7, 15, 1, 17, 9, 197, 49, 51, 127, 11, 577, 35, 1, 37, 721, 13, 199, 24335, 97, 99, 649, 485, 15, 19603, 31, 63, 1, 65, 33, 251, 17, 3699, 57799, 53, 161, 163, 55, 10405, 77617, 19, 1151, 2143295, 4801, 99, 1, 101, 5201, 32080051, 1351, 21, 127
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(1), 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(x))};