A067614 - OEIS

A067614

a(n) is the second partial quotient in the simple continued fraction for sqrt(prime(n)).

2, 1, 4, 1, 3, 1, 8, 2, 1, 2, 1, 12, 2, 1, 1, 3, 1, 1, 5, 2, 1, 1, 9, 2, 1, 20, 6, 2, 2, 1, 3, 2, 1, 1, 4, 3, 1, 1, 1, 6, 2, 2, 1, 1, 28, 9, 1, 1, 15, 7, 3, 2, 1, 1, 32, 4, 2, 2, 1, 1, 1, 8, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 6, 3, 2, 1, 1, 1, 40, 4, 2, 1, 1, 1, 1, 21, 5, 2, 2, 1, 1, 1, 14, 6, 2, 2, 1, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..99.

FORMULA

a(n) = floor(1/(sqrt(prime(n))-floor(sqrt(prime(n))))), where prime(n) is the n-th prime.

a(n) = floor(2*s/r) where s = floor(sqrt(p)) = A000006(n), r = p - s^2 = A056892(n), and p = prime(n). - Kevin Ryde, May 06 2022

EXAMPLE

For n=8, prime(n)=19, floor(sqrt(19))=4 and 1/(sqrt(19)-4) = 2.786..., so a(8)=2.

MATHEMATICA

a[n_] := Floor[1/(Sqrt[Prime[n]]-Floor[Sqrt[Prime[n]]])]