A003814 Numbers k such that the continued fraction for sqrt(k) has odd period length.

2, 5, 10, 13, 17, 26, 29, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 149, 157, 170, 173, 181, 185, 193, 197, 202, 218, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 290, 293, 298, 313, 314, 317

Offset: 1

Author

N. J. A. Sloane, Walter Gilbert

Keywords

• nonn

Comments

All primes of the form 4m + 1 are here. - T. D. Noe, Mar 19 2012

These numbers have no prime factors of the form 4m + 3. - Thomas Ordowski, Jul 01 2013

This sequence is a proper subsequence of the so-called 1-happy number products A007969. See the W. Lang link there, eq. (1), with B = 1, C = a(n), also with a table at the end. This is due to the soluble Pell equation R^2 - C*S^2 = -1 for C = a(n). See e.g., Perron, Satz 3.18. on p. 93, and the table on p. 91 with the numbers D of the first column that do not have a number in brackets in the second column (Teilnenner von sqrt(D)). - Wolfdieter Lang, Sep 19 2015

References

• W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
• O. Perron, Die Lehre von den Kettenbrüchen, Band I, Teubner Verlagsgesellschaft, Stuttgart, 1954.
• Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• S. R. Finch, Class number theory [Cached copy, with permission of the author]
• P. J. Rippon and H. Taylor, Even and odd periods in continued fractions of square roots, Fibonacci Quarterly 42, May 2004, pp. 170-180.

Crossrefs

Cf. A010333, A003654, A007969.

Cf. A031396.

Cf. A206586 (period has positive even length).

Programs

Maple

isA003814 := proc(n)
    local cf,p ;
    if issqr(n) then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        if modp(p,4) = 3 then
            return false;
        end if;
    end do:
    cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;
    type( nops(op(2,cf)),'odd') ;
end proc:
A003814 := proc(n)
    option remember;
    if n = 1 then
        2;
    else
        for a from procname(n-1)+1 do
            if isA003814(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A003814(n),n=1..40) ; # R. J. Mathar, Oct 19 2014

Mathematica

Select[Range[100], ! IntegerQ[Sqrt[#]] && OddQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &] (* T. D. Noe, Mar 19 2012 *)

PARI

cyc(cf) = {
  if(#cf==1, return([])); \\ There is no cycle
  my(s=[]);
  for(k=2, #cf,
    s=concat(s, cf[k]);
    if(cf[k]==2*cf[1], return(s)) \\ Cycle found
  );
  0 \\ Cycle not found
}
select(n->#cyc(contfrac(sqrt(n)))%2==1, vector(400, n, n)) \\ Colin Barker, Oct 19 2014