A338785 - OEIS

%I #11 Nov 11 2020 19:37:57

%S 1,2,13,4,6,8,21,11,30,14,18,27,44,41,29,43,37,34,68,36,42,94,147,58,

%T 88,47,186,93,142,75,110,90,112,67,178,228,82,114,100,222,187,105,191,

%U 143,204,131,180,115,172,177,197,133,263,272,353,175,231,242,322,157

%N a(n) is the least number k such that continued fraction for sqrt(prime(k)) has period n.

%H Robert Israel, <a href="/A338785/b338785.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A000720(A059800(n)).

%e sqrt(prime(1))  = sqrt(2)  = 1 + 1/(2 + 1/(2 + ...)), period 1.

%e sqrt(prime(2))  = sqrt(3)  = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...)))), period 2.

%e sqrt(prime(13)) = sqrt(41) = 6 + 1/(2 + 1/(2 + 1/(12 + 1/(2 + 1/(2 + 1/(12 + ...)))))), period 3.

%p N:= 100: # for a(1)..a(N)

%p A:= Vector(N): count:= 0: p:= 1:

%p for n from 1 while count < N do

%p   p:= nextprime(p);

%p   v:= nops(numtheory:-cfrac(sqrt(p),periodic,quotients)[2]);

%p   if v <= N and A[v] = 0 then count:= count+1; A[v]:= n; fi

%p od:

%p convert(A,list); # _Robert Israel_, Nov 11 2020

%t Table[SelectFirst[Range[500], Length[Last[ContinuedFraction[Sqrt[Prime[#]]]]] == n &], {n, 60}]

%Y Cf. A000720, A013646, A054269, A059800.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Nov 08 2020