A308780 - OEIS

%I #16 May 04 2024 14:57:49

%S 1,2,1,3,2,1,4,2,1,5,2,1,6,4,3,2,1,7,2,1,8,4,2,1,9,6,3,2,1,10,5,4,2,1,

%T 11,2,1,12,8,6,4,3,2,1,13,2,1,14,7,4,2,1,15,10,6,5,3,2,1,16,8,4,2,1,

%U 17,2,1,18,12,9,6,4,3,2,1,19,2,1

%N First element of the periodic part of the continued fraction expansion of sqrt(k), where the period is 2.

%H Georg Fischer, <a href="https://github.com/gfis/fasces/blob/master/oeis/cfsqrt/sqrt20k.txt">Table of the continued fractions of sqrt(0..20000)</a>.

%e The continued fractions for sqrt(3..8) are:

%e    3 1;1,2

%e    4 2 (square)

%e    5 2;4

%e    6 2;2,4

%e    7 2;1,1,1,4

%e    8 2;1,4

%e Those for 3, 6 and 8 have a period of 2, therefore the sequence starts with 1, 2, 1.

%p s := proc(n) if not issqr(n) then numtheory[cfrac](sqrt(n), 'periodic', 'quotients')[2]; if nops(%) = 2 then return %[1] fi fi; NULL end:

%p seq(s(n), n=1..399); # _Peter Luschny_, Jul 01 2019

%t Reap[For[k = 3, k <= 399, k++, If[!IntegerQ[Sqrt[k]], cf = ContinuedFraction[Sqrt[k]]; If[Length[cf[[2]]] == 2, Sow[cf[[2, 1]]]]]]][[2, 1]] (* _Jean-François Alcover_, May 03 2024 *)

%t (* Second program (much simpler): *)

%t Table[2 a/b, {a, 1, 20}, {b, Rest@Divisors[2 a]}] // Flatten (* _Jean-François Alcover_, May 04 2024, after a remark by _Kevin Ryde_ *)

%Y Cf. A013642, A308778.

%K nonn

%O 1,2

%A _Georg Fischer_, Jun 24 2019