OFFSET
1,1
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Walter Gilbert
STATUS
approved