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).

0, 1, 2, 3, 17, 31, 45, 138, 231, 617, 72496, 144375, 216254, 288133, 360012, 431891, 503770, 575649, 647528, 719407, 791286, 863165, 935044, 1006923, 1078802, 1150681, 1222560, 1294439, 1366318, 1438197, 1510076, 1581955, 1653834, 1725713, 1797592, 1869471, 1941350, 2013229, 2085108, 2156987

Offset: 1

Author

V. Raman, Oct 11 2012

Keywords

• nonn
• base

Comments

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

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.

Crossrefs

Cf. A217684, A217685, A217686, A217687.

Programs

PARI

default(realprecision, 1000); a=vector(100,i,(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , i))[2, 1]))
log_fibonacci(j)=(j*log((1+sqrt(5))/2)/log(10))-(log(sqrt(5))/log(10))
deviation(k)=abs(round(log_fibonacci(k))-log_fibonacci(k))
n=6;err=deviation(n);m=3;while(n<10^20,if(deviation(n+a[m])