A217687 Values of n such that Fibonacci(n) gets increasingly closer to the powers of 10 (measured by the ratio between the Fibonacci number and the nearest power of 10).

1, 2, 6, 11, 16, 83, 150, 217, 662, 1107, 2954, 346893, 690832, 1034771, 1378710, 1722649, 2066588, 2410527, 2754466, 3098405, 3442344, 3786283, 4130222, 4474161, 4818100, 5162039, 5505978, 5849917, 6193856, 6537795, 6881734, 7225673, 7569612, 7913551, 8257490, 8601429, 8945368, 9289307, 9633246, 9977185

Offset: 0

Author

V. Raman, Oct 11 2012

Keywords

• nonn
• base

Comments

The sequence A217686 gives the sequence of values n such that Lucas(n) get increasingly closer to the powers of 10 (by the ratio between the Lucas number to the nearest power of 10).

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 A217686.

Crossrefs

Cf. A217684, A217685, A217686, A217688.

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])