A233588 Decimal expansion of the continued fraction prime(1) + prime(1)/(prime(2) + prime(2)/(prime(3) + prime(3)/(prime(4) + prime(4)/...))).

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

Offset

1,1

Comments

Given a nondecreasing sequence s(n) of natural numbers (such as, in this case, that of primes A000079), the corresponding continued fraction is bf(s(n)) = a(1) + a(1)/(a(2) + a(2)/(a(3) + a(3)/(...))).

For the inverse of this mapping of nondecreasing sequences of natural numbers into irrational real numbers greater than 1, see A233582.

Links

Stanislav Sykora, Table of n, a(n) for n = 1..25000 [a(224) onward corrected by Kevin Ryde, Oct 27 2024]

Stanislav Sykora, Blazys' Expansions and Continued Fractions, Stans Library, Vol.IV, 2013, DOI 10.3247/sl4math13.001

Stanislav Sykora, PARI/GP scripts for Blazys expansions and fractions, OEIS Wiki

Formula

Equals 2 + 2/(3 + 3/(5 + 5/(7 + 7/(11 + 11/(13 + 13/(17 + ...)))))).

Example

2.56654383217138884446752910633228575178297282870231464596973352546639971...

Mathematica

RealDigits[ Fold[(#2 + #2/#1) &, 1, Reverse@ Prime@ Range@ 57], 10, 111][[1]] (* Robert G. Wilson v, May 22 2014 *)

Prog

(PARI) See the link.

Crossrefs

Cf. A000079 (primes), A233589, A233590, A233591, A233582, A233583, A233584, A233585, A233586, A233587.

Sequence in context: A155947 A008294 A019694 * A113975 A035585 A239868

Adjacent sequences: A233585 A233586 A233587 * A233589 A233590 A233591

Keyword

nonn,cons

Author

Stanislav Sykora, Jan 02 2014

Status

approved