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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
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
0,2
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.
PROG
(PARI) default(realprecision, 1000); a=vector(100, i, (contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), 0, 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])<err, n=n+a[m]; err=deviation(n); print(n), m++))
CROSSREFS
KEYWORD
nonn,base
AUTHOR
V. Raman, Oct 11 2012
STATUS
approved