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

1,2

Links

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

Formula

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

Example

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.

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}]

Crossrefs

Cf. A000720, A013646, A054269, A059800.

Sequence in context: A286459 A358433 A344540 * A335817 A093079 A095417

Adjacent sequences: A338782 A338783 A338784 * A338786 A338787 A338788

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 08 2020

Status

approved