|
%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
|
