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

Offset: 1

Author

Roger L. Bagula, Feb 01 2002

Keywords

• nonn
• easy

Examples

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

Crossrefs

Cf. A000006, A000040, A056892.

Programs

Mathematica

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

PARI

a(n) = my(r); sqrtint(prime(n),&r)<<1 \ r; \\ Kevin Ryde, May 06 2022

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

Extensions

Edited by Dean Hickerson, Feb 14 2002