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

1,1

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]]])]

Prog

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

Crossrefs

Cf. A000006, A000040, A056892.

Sequence in context: A329900 A101261 A214057 * A341308 A353359 A113901

Adjacent sequences: A067611 A067612 A067613 * A067615 A067616 A067617

Keyword

nonn,easy

Author

Roger L. Bagula, Feb 01 2002

Extensions

Edited by Dean Hickerson, Feb 14 2002

Status

approved