%I #14 Jan 01 2022 09:56:55

%S 11,81,765,7581,75703,756903,7568866,75688472,756884504,7568844796,

%T 75688447681,756884476508,7568844764750,75688447647137,

%U 756884476470980,7568844764709381,75688447647093366,756884476470933182

%N 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.

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

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

%e (80+1)/2 - (80!)^(1/80) = 9.9026... < 10^1 < 10.032... = (81+1)/2 - (81!)^(1/81)

%e So 81 is the smallest k where the required difference exceeds 10, thus a(1) = 81.

%o (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

%K nonn

%O 0,1

%A _Stefan Steinerberger_, Nov 05 2005

%E More terms from _Robert Gerbicz_, Aug 24 2006