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A003814
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Numbers k such that the continued fraction for sqrt(k) has odd period length.
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50
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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!).
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LINKS
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MAPLE
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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:
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:
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MATHEMATICA
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Select[Range[100], ! IntegerQ[Sqrt[#]] && OddQ[Length[ContinuedFraction[Sqrt[#]][[2]]]] &] (* T. D. Noe, Mar 19 2012 *)
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PROG
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(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
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CROSSREFS
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Cf. A206586 (period has positive even length).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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