A077426 - OEIS

%I #25 Aug 07 2015 02:41:54

%S 5,13,17,29,37,41,53,61,65,73,85,89,97,101,109,113,125,137,145,149,

%T 157,173,181,185,193,197,229,233,241,257,265,269,277,281,293,313,317,

%U 325,337,349,353,365,373,389,397,401,409,421,425,433,445,449,457,461,481,485

%N Nonsquare integers n such that the continued fraction (sqrt(n)+1)/2 has odd period length.

%C Nonsquare integers n for which Pell equation x^2 - n*y^2 = -4 has infinitely many integer solutions. The smallest solutions are given in A078356 and A078357.

%D O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, table p. 108).

%H Vincenzo Librandi, <a href="/A077426/b077426.txt">Table of n, a(n) for n = 1..1000</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pell%27s_equation">Pell's equation</a>

%p isOddPrim := proc(n::integer)

%p     local cf;

%p     cf := numtheory[cfrac]((sqrt(n)+1)/2,'periodic','quotients') ;

%p     if nops(op(2,cf)) mod 2 =1 then

%p         RETURN(true) ;

%p     else

%p         RETURN(false) ;

%p     fi ;

%p end:

%p notA077426 := proc(n::integer)

%p     if issqr(n) then

%p         RETURN(true) ;

%p     elif not isOddPrim(n) then

%p         RETURN(true) ;

%p     else

%p         RETURN(false) ;

%p     fi ;

%p end:

%p A077426 := proc(n::integer)

%p     local resul,i ;

%p     resul := 5 ;

%p     i := 1 ;

%p     while i < n do

%p         resul := resul+4 ;

%p         while notA077426(resul) do

%p             resul := resul+4 ;

%p         od ;

%p         i:= i+1 ;

%p     od ;

%p     RETURN(resul) ;

%p end:

%p for n from 1 to 61 do print(A077426(n)) ; od : # _R. J. Mathar_, Apr 25 2006

%t fQ[n_] := !IntegerQ@ Sqrt@ n && OddQ@ Length@ ContinuedFraction[(Sqrt@ n + 1)/2][[2]]; Select[Range@ 500, fQ] (* _Robert G. Wilson v_, Nov 17 2012 *)

%Y A subsequence of A077425.

%Y Odd elements of A003814.

%Y Cf. A077427, A172000.

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_, Nov 29 2002

%E Edited and extended by _Max Alekseyev_, Mar 03 2010

%E Edited by _Max Alekseyev_, Mar 05 2010