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A217688 Values of n such that 10^n gets increasingly closer to a Fibonacci number (measured by the ratio between the power of 10 and the nearest Fibonacci number). 1

%I

%S 0,1,2,3,17,31,45,138,231,617,72496,144375,216254,288133,360012,

%T 431891,503770,575649,647528,719407,791286,863165,935044,1006923,

%U 1078802,1150681,1222560,1294439,1366318,1438197,1510076,1581955,1653834,1725713,1797592,1869471,1941350,2013229,2085108,2156987

%N Values of n such that 10^n gets increasingly closer to a Fibonacci number (measured by the ratio between the power of 10 and the nearest Fibonacci number).

%C The sequence A217685 gives the sequence of values n such that 10^n gets increasingly closer to a Lucas number.

%C Given that for sufficiently large values of n, Fibonacci(n) ~ Lucas(n)/sqrt(5) ~ (((1+sqrt(5))/2)^n)/(sqrt(5)), the intermediate differences between the terms in this sequence also need to be a member of the sequence A217685.

%o (PARI) default(realprecision, 1000); a=vector(100,i,(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), 0, i))[2, 1]))

%o log_fibonacci(j)=(j*log((1+sqrt(5))/2)/log(10))-(log(sqrt(5))/log(10))

%o deviation(k)=abs(round(log_fibonacci(k))-log_fibonacci(k))

%o n=6;err=deviation(n);m=3;while(n<10^20,if(deviation(n+a[m])<err,n=n+a[m];err=deviation(n);print(round(log_fibonacci(n))),m++))

%Y Cf. A217684, A217685, A217686, A217687.

%K nonn,base

%O 1,3

%A _V. Raman_, Oct 11 2012

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Last modified November 29 14:50 EST 2021. Contains 349416 sequences. (Running on oeis4.)