A064442 Decimal expansion of number with continued fraction expansion 2, 3, 5, 7, 11, 13, 17, 19, ... = 2.3130367364335829063839516 ...

2, 3, 1, 3, 0, 3, 6, 7, 3, 6, 4, 3, 3, 5, 8, 2, 9, 0, 6, 3, 8, 3, 9, 5, 1, 6, 0, 2, 6, 4, 1, 7, 8, 2, 4, 7, 6, 3, 9, 6, 6, 8, 9, 7, 7, 1, 8, 0, 3, 2, 5, 6, 3, 4, 0, 2, 1, 0, 1, 2, 4, 4, 4, 2, 1, 4, 4, 5, 6, 4, 7, 3, 1, 7, 7, 6, 2, 7, 2, 2, 4, 3, 6, 9, 5, 3, 2, 2, 0, 1, 7, 2, 3, 8, 3, 2, 8, 1, 7, 4, 5, 3, 0, 1, 5, 8, 2

Offset

1,1

Comments

Continued fraction expansion of the prime numbers. - Harvey P. Dale, Sep 25 2012

Links

Table of n, a(n) for n=1..107.

MathOverflow, What is the effect of adding 1/2 to a continued fraction?

Formula

1/A084255. - Franklin T. Adams-Watters, Jul 31 2009

From Peter Bala, Nov 26 2019: (Start)

Denoting the constant by c we have the related simple continued fraction expansions (prime(n) denotes the n-th prime number):

2*c = [4; 1, 1, 1, 2, 14, 5, 1, 1, 6, 34, 9, 1, 1, 11, 58, 15, 1, 1, 18, 82, 21, ..., 1, 1, (prime(3*n) - 1)/2, 2*prime(3*n+1), (prime(3*n+2) - 1)/2, ...];

(1/2)*c = [1; 6, 2, 1, 1, 3, 22, 6, 1, 1, 8, 38, 11, 1, 1, 14, 62, 18, 1, 1, 20, 86, 23, ..., 1, 1, (prime(3*n+1) - 1)/2, 2*prime(3*n+2), (prime(3*n+3) - 1)/2, ...];

(c + 1)/(c - 1) = [2; 1, 1, 10, 3, 1, 1, 5, 26, 8, 1, 1, 9, 46, 14, 1, 1, 15, 74, 20, ..., 1, 1, (prime(3*n+2) - 1)/2, 2*prime(3*n+3), (prime(3*n+4) - 1)/2, ...]. (End)

Example

2.313036736433582906383951602641782476396689771803256340210124442144564731776...

Mathematica

RealDigits[ N[ FromContinuedFraction[ Table[ Prime[n], {n, 1, 100} ]], 100]] [[1]]
RealDigits[FromContinuedFraction[Prime[Range[200]]], 10, 120][[1]] (* Harvey P. Dale, Sep 06 2021 *)

Crossrefs

Cf. A060997, A302937.

Sequence in context: A082503 A190549 A353640 * A287566 A134411 A126044

Adjacent sequences: A064439 A064440 A064441 * A064443 A064444 A064445

Keyword

cons,nonn

Author

Robert G. Wilson v, Oct 01 2001

Status

approved