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A073833 Numerators of b(n) where b(1) = 1, b(i) = b(i-1) + 1/b(i-1). 5
1, 2, 5, 29, 941, 969581, 1014556267661, 1099331737522548368039021, 1280590510388959061548230114212510564911731118541, 1726999038066943724857508638586386504281539279376091034086485112150121338989977841573308941492781 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is also the numerator of the fractional chromatic number of the Mycielski graph M_n. - Eric W. Weisstein, Mar 05 2011

It appears that lim n -> infinity (1/n)*exp(2*(b(n)^2-2n)) = c1 = 0.57...... - Benoit Cloitre, Oct 16 2002

c1 = 0.574810274671785...; see A232975. - Jon E. Schoenfield, Nov 30 2013

b(n)^2 = t/2 + u + (u-1/2)/t + (-u^2+2*u-11/12)/t^2 + (4*u^3/3-5*u^2+17*u/3-65/36)/t^3 + ... where t=4*n, u=(log n)/2+c, and c=-0.2768576248625765389364372...; see A233770. - Jon E. Schoenfield, Dec 15 2013

REFERENCES

H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 187.

D. J. Newman, A Problem Seminar, Springer; see Problem #60.

J. H. Silverman, The arithmetic of dynamical systems, Springer, 2007, see p. 113 Table 3.1

LINKS

Table of n, a(n) for n=1..10.

Eric Weisstein's World of Mathematics, Fractional Chromatic Number

FORMULA

a(n) = a(n-1)^2 + A073834(n-1)^2; A073834(n) = a(n-1) * A073834(n-1). - Franklin T. Adams-Watters, Aug 04 2008

EXAMPLE

1, 2, 5/2, 29/10, 941/290, 969581/272890, 1014556267661/264588959090, 1099331737522548368039021/268440386798659418988490, ...

MATHEMATICA

f[n_]:=n+1/n; Prepend[Numerator[NestList[f, 2, 9]], 1] [Vladimir Joseph Stephan Orlovsky, Nov 19 2010]

Numerator[NestList[# + 1/# &, 1, 10]] [Eric W. Weisstein, Mar 05 2001]

PROG

(PARI) {a(n) = local(x, y); if( n<1, 0, if( n<3, n, x = a(n-2)^2; y = a(n-1); y^2 + x * (y - x)))} /* Michael Somos, Mar 05 2012 */

CROSSREFS

See A073834 for denominators. See A232975 for c1; see A233770 for c.

Sequence in context: A059784 A000283 A121910 * A229918 A179554 A086383

Adjacent sequences:  A073830 A073831 A073832 * A073834 A073835 A073836

KEYWORD

frac,nonn

AUTHOR

Alex Fink (finks(AT)telus.net), Aug 12 2002

STATUS

approved

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Last modified July 30 17:25 EDT 2014. Contains 245072 sequences.