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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

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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

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

S. 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