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A338785
a(n) is the least number k such that continued fraction for sqrt(prime(k)) has period n.
1
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 08 2020
STATUS
approved