A077426 Nonsquare integers n such that the continued fraction (sqrt(n)+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

Author

Wolfdieter Lang, Nov 29 2002

Keywords

• nonn
• easy

Comments

Nonsquare integers n for which Pell equation x^2 - n*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

Crossrefs

A subsequence of A077425.

Odd elements of A003814.

Cf. A077427, A172000.

Programs

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 *)

Extensions

Edited and extended by Max Alekseyev, Mar 03 2010

Edited by Max Alekseyev, Mar 05 2010