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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).

%I A217687 #14 Aug 07 2025 19:20:41
%S A217687 1,2,6,11,16,83,150,217,662,1107,2954,346893,690832,1034771,1378710,
%T A217687 1722649,2066588,2410527,2754466,3098405,3442344,3786283,4130222,
%U A217687 4474161,4818100,5162039,5505978,5849917,6193856,6537795,6881734,7225673,7569612,7913551,8257490,8601429,8945368,9289307,9633246,9977185
%N 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).
%C A217687 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).
%C A217687 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.
%o A217687 (PARI) default(realprecision, 1000); a=vector(100,i,(contfracpnqn(contfrac(log((1+sqrt(5))/2)/log(10), , i))[2, 1]))
%o A217687 log_fibonacci(j)=(j*log((1+sqrt(5))/2)/log(10))-(log(sqrt(5))/log(10))
%o A217687 deviation(k)=abs(round(log_fibonacci(k))-log_fibonacci(k))
%o A217687 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++))
%Y A217687 Cf. A217684, A217685, A217686, A217688.
%K A217687 nonn,base
%O A217687 0,2
%A A217687 _V. Raman_, Oct 11 2012