A111334 a(n) is the smallest integer k such that the difference between the arithmetic and geometric means of the first k positive integers is larger than 10^n.

11, 81, 765, 7581, 75703, 756903, 7568866, 75688472, 756884504, 7568844796, 75688447681, 756884476508, 7568844764750, 75688447647137, 756884476470980, 7568844764709381, 75688447647093366, 756884476470933182

Offset

0,1

Comments

By using the approximation formula k! = (k/e)^k one can show that a(n) will be approximately 7.56*10^n.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Min_{k: (k+1)/2 - (k!)^(1/k) > 10^n}.

Example

(80+1)/2 - (80!)^(1/80) = 9.9026... < 10^1 < 10.032... = (81+1)/2 - (81!)^(1/81)
So 81 is the smallest k where the required difference exceeds 10, thus a(1) = 81.

Prog

(PARI) f(n)=return(log(sqrt(2*Pi))+(n+0.5)*log(n)-n+1/(12*n)) for(k=0, 24, n=0; forstep(i=4*k+8, 0, -1, m=n+2^i; \ if(f(m)>m*log((m+1)/2-10^k), n=m)); print1(n+1, ", ")) \\ Robert Gerbicz, Aug 24 2006

Crossrefs

Sequence in context: A003730 A334340 A335332 * A085879 A239459 A197643

Adjacent sequences: A111331 A111332 A111333 * A111335 A111336 A111337

Keyword

nonn

Author

Stefan Steinerberger, Nov 05 2005

Extensions

More terms from Robert Gerbicz, Aug 24 2006

Status

approved