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

1,1

Links

Table of n, a(n) for n=1..56.

Carlos Rivera, Problem 88. Follow-up to problem 63, The Prime Puzzles & Problems Connection.

Formula

a(n) = A002350(8*n).

a(n) = sqrt(8*n*A368339(n)^2 + 1).

a(A000217(n)) = 2*n + 1, n >= 1.

Example

For n = 1, 2, 3, 4, 5 solutions are (x,y) = (3, 1), (1, 0), (5, 1), (17, 3), (19, 3).

Prog

(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))};

Crossrefs

Cf. A000217, A002350, A069129, A368339.

Sequence in context: A124740 A369121 A347556 * A338172 A104053 A187369

Adjacent sequences: A368337 A368338 A368339 * A368341 A368342 A368343

Keyword

nonn,easy

Author

Arkadiusz Wesolowski, Dec 21 2023

Status

approved