A338785 a(n) is the least number k such that continued fraction for sqrt(prime(k)) has period n.

1, 2, 13, 4, 6, 8, 21, 11, 30, 14, 18, 27, 44, 41, 29, 43, 37, 34, 68, 36, 42, 94, 147, 58, 88, 47, 186, 93, 142, 75, 110, 90, 112, 67, 178, 228, 82, 114, 100, 222, 187, 105, 191, 143, 204, 131, 180, 115, 172, 177, 197, 133, 263, 272, 353, 175, 231, 242, 322, 157

Offset: 1

Author

Ilya Gutkovskiy, Nov 08 2020

Keywords

• nonn

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000720, A013646, A054269, A059800.

Programs

Maple

N:= 100: # for a(1)..a(N)
A:= Vector(N): count:= 0: p:= 1:
for n from 1 while count < N do
  p:= nextprime(p);
  v:= nops(numtheory:-cfrac(sqrt(p),periodic,quotients)[2]);
  if v <= N and A[v] = 0 then count:= count+1; A[v]:= n; fi
od:
convert(A,list); # Robert Israel, Nov 11 2020

Mathematica

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

Formula

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