A003814 - OEIS

%I #66 Oct 29 2023 01:40:34

%S 2,5,10,13,17,26,29,37,41,50,53,58,61,65,73,74,82,85,89,97,101,106,

%T 109,113,122,125,130,137,145,149,157,170,173,181,185,193,197,202,218,

%U 226,229,233,241,250,257,265,269,274,277,281,290,293,298,313,314,317

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

%C All primes of the form 4m + 1 are here. - _T. D. Noe_, Mar 19 2012

%C These numbers have no prime factors of the form 4m + 3. - _Thomas Ordowski_, Jul 01 2013

%C 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

%D W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.

%D O. Perron, Die Lehre von den Kettenbrüchen, Band I, Teubner Verlagsgesellschaft, Stuttgart, 1954.

%D Kenneth H. Rosen, Elementary Number Theory and Its Applications, Addison-Wesley, 1984, page 426 (but beware of errors!).

%H Ray Chandler, <a href="/A003814/b003814.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)

%H S. R. Finch, <a href="/A000924/a000924.pdf">Class number theory</a> [Cached copy, with permission of the author]

%H P. J. Rippon and H. Taylor, <a href="http://www.fq.math.ca/Papers1/42-2/quartrippon02_2004.pdf">Even and odd periods in continued fractions of square roots</a>, Fibonacci Quarterly 42, May 2004, pp. 170-180.

%p isA003814 := proc(n)

%p     local cf,p ;

%p     if issqr(n) then

%p         return false;

%p     end if;

%p     for p in numtheory[factorset](n) do

%p         if modp(p,4) = 3 then

%p             return false;

%p         end if;

%p     end do:

%p     cf := numtheory[cfrac](sqrt(n),'periodic','quotients') ;

%p     type( nops(op(2,cf)),'odd') ;

%p end proc:

%p A003814 := proc(n)

%p     option remember;

%p     if n = 1 then

%p         2;

%p     else

%p         for a from procname(n-1)+1 do

%p             if isA003814(a) then

%p                 return a;

%p             end if;

%p         end do:

%p     end if;

%p end proc:

%p seq(A003814(n),n=1..40) ; # _R. J. Mathar_, Oct 19 2014

%t Select[Range[100], ! IntegerQ[Sqrt[#]] && OddQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &] (* _T. D. Noe_, Mar 19 2012 *)

%o (PARI)

%o cyc(cf) = {

%o   if(#cf==1, return([])); \\ There is no cycle

%o   my(s=[]);

%o   for(k=2, #cf,

%o     s=concat(s, cf[k]);

%o     if(cf[k]==2*cf[1], return(s)) \\ Cycle found

%o   );

%o   0 \\ Cycle not found

%o }

%o select(n->#cyc(contfrac(sqrt(n)))%2==1, vector(400, n, n)) \\ _Colin Barker_, Oct 19 2014

%Y Cf. A010333, A003654, A007969.

%Y Cf. A031396.

%Y Cf. A206586 (period has positive even length).

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Walter Gilbert