A191504 Decimal expansion of the number 1/(1+1/(1+2/(1+3/(1+5/(1+7/(1+11/(1+13/(1+17/(1+19/(1+... )))))))))), where coefficients > 1 are the primes.

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

Offset: 0

Author

Fabrice Auzanneau, Jun 04 2011

Keywords

• nonn
• cons

Comments

The number can be written 1/(1+s(0)) with s(k)=prime(k)/(1+s(k+1)), prime(0):=1. Asymptotically, s(k) ~ sqrt(prime(k)).

Examples

			0.6620942517851037588123181089841636860733854770812446632320193128554043...
		

Crossrefs

Cf. A191608.

Programs

Mathematica

N[Fold[#2/(1 + #1) &, 0, Join[Reverse@Prime@Range@180000, {1, 1}]], 111] (* Robert G. Wilson v, Jun 16 2011 *)

PARI

default(realprecision,80); s=sqrt(p=1e6); while(p=precprime(p-1),s=p/(1+s)); eval(vecextract(Vec(Str((1+s)/(2+s))),"3..-2"))  \\ M. F. Hasler, Jun 16 2011

Formula

1/(1+1/(1+2/(1+3/(1+5/(1+7/(1+11/(1+13/(1+17/(1+19/(1+... ))))))))))

Extensions

Values corrected upon observation by R. J. Mathar, Jun 16 2011

Corrected and extended by Max Alekseyev, Aug 11 2013