OFFSET
1,1
COMMENTS
Together with {1} and A031398 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - Max Alekseyev, Mar 09 2010
Squarefree integers m such that Q(sqrt(m)) contains the infinite continued fraction [k, k, k, k, k, ...] for some positive integer k. For example, Q(sqrt(5)) contains [1, 1, 1, 1, 1, ...] which equals (1 + sqrt(5))/2. - Greg Dresden, Jul 23 2010
REFERENCES
D. A. Buell, Binary Quadratic Forms. Springer-Verlag, NY, 1989, pp. 224-241.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 46.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 56.
W. Paulsen, Calkin-Wilf sequences for irrational numbers, Fib. Q., 61:1 (2023), 51-59.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
R. J. Mathar, Table of n, a(n) for n = 1..9446
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
P. Seeling, Ueber die Aufloesung der Gleichung x^2-Ay^2=+-1 in ganzen Zahlen, wo A positiv und kein vollstaendiges Quadrat sein muss, Archiv der Mathematik und Physik, Vol. 52 (1871), p. 40-49.
MAPLE
isA003654 := proc(n)
local cf, p ;
if not numtheory[issqrfree](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:
A003654 := proc(n)
option remember;
local a;
if n = 1 then
2;
else
for a from procname(n-1)+1 do
if isA003654(a) then
return a;
end if;
end do:
end if;
end proc:
seq(A003654(n), n=1..40) ; # R. J. Mathar, Oct 19 2014
MATHEMATICA
Reap[For[n = 2, n < 1000, n++, If[SquareFreeQ[n], sol = Solve[x^2 - n y^2 == -1, {x, y}, Integers]; If[sol != {}, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Mar 24 2020 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Max Alekseyev, Mar 17 2010
STATUS
approved