A077426 Nonsquare integers k such that the continued fraction (sqrt(k)+1)/2 has odd period length.

5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 173, 181, 185, 193, 197, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 389, 397, 401, 409, 421, 425, 433, 445, 449, 457, 461, 481, 485

Offset

1,1

Comments

Nonsquare integers k for which Pell equation x^2 - k*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.

References

O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Wikipedia, Pell's equation

Maple

isOddPrim := proc(n::integer)
    local cf;
    cf := numtheory[cfrac]((sqrt(n)+1)/2, 'periodic', 'quotients') ;
    if nops(op(2, cf)) mod 2 =1 then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
notA077426 := proc(n::integer)
    if issqr(n) then
        RETURN(true) ;
    elif not isOddPrim(n) then
        RETURN(true) ;
    else
        RETURN(false) ;
    fi ;
end:
A077426 := proc(n::integer)
    local resul, i ;
    resul := 5 ;
    i := 1 ;
    while i < n do
        resul := resul+4 ;
        while notA077426(resul) do
            resul := resul+4 ;
        od ;
        i:= i+1 ;
    od ;
    RETURN(resul) ;
end:
for n from 1 to 61 do print(A077426(n)) ; od : # R. J. Mathar, Apr 25 2006

Mathematica

fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* Robert G. Wilson v, Nov 17 2012 *)

Prog

(Python)
from itertools import count
from sympy.solvers.diophantine.diophantine import diop_DN
def A077426_gen(startvalue=3): # generator of terms >= startvalue
    return filter(lambda n: len(diop_DN(n, -1)), count(max(startvalue+(startvalue&1^1), 3), 2)) # Chai Wah Wu, Dec 21 2025

Crossrefs

A subsequence of A077425.

Odd elements of A003814.

Cf. A077427, A172000.

Sequence in context: A359151 A191218 A279857 * A231754 A387595 A175768

Adjacent sequences: A077423 A077424 A077425 * A077427 A077428 A077429

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 29 2002

Extensions

Edited and extended by Max Alekseyev, Mar 03 2010

Edited by Max Alekseyev, Mar 05 2010

Status

approved