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A072353 a(n) is the index of the largest Fibonacci number containing n digits.

%I A072353 #41 Jul 13 2025 19:53:00
%S A072353 6,11,16,20,25,30,35,39,44,49,54,59,63,68,73,78,83,87,92,97,102,106,
%T A072353 111,116,121,126,130,135,140,145,150,154,159,164,169,173,178,183,188,
%U A072353 193,197,202,207,212,216,221,226,231,236,240,245,250,255,260,264,269,274
%N A072353 a(n) is the index of the largest Fibonacci number containing n digits.
%C A072353 Partial sums of A050815: a(n) = Sum_{k=1..n} A050815(k). - _Reinhard Zumkeller_, Apr 14 2005
%C A072353 Equivalently, a(n) is the number of Fibonacci numbers < 10^n including F(0) = 0 and F(1) = F(2) = 1 once. - _Derek Orr_, Jun 01 2014
%H A072353 Harvey P. Dale, <a href="/A072353/b072353.txt">Table of n, a(n) for n = 1..1000</a>
%H A072353 Jürgen Spilker, <a href="http://dx.doi.org/10.5169/seals-8480">Die Ziffern der Fibonacci-Zahlen</a>, Elemente der Mathematik 58 (Birkhäuser 2003).
%H A072353 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%F A072353 Limit_{n->oo} a(n)/n = 1/log_10((1+sqrt(5))/2) = 1/A097348 = 4.784... . - _Reinhard Zumkeller_, Apr 14 2005.
%F A072353 a(n) = floor(n*log(10)/log(phi)+log(5)/(2*log(phi))), where phi=(1+sqrt(5))/2, the golden ratio. - _Hans J. H. Tuenter_, Jul 08 2025
%e A072353 a(3)=16, as the 16th Fibonacci number is the largest Fibonacci number with 3 digits.
%t A072353 With[{fibs=Fibonacci[Range[300]]},Flatten[Position[fibs,#]&/@ Table[ Max[ Select[fibs,IntegerLength[#]==n&]],{n,60}]]] (* _Harvey P. Dale_, Nov 09 2011 *)
%o A072353 (Python)
%o A072353 def A072353_list(n):
%o A072353     list = []
%o A072353     x, y, index = 1, 1, 1
%o A072353     while len(list) < n:
%o A072353         if len(str(x)) < len(str(y)):
%o A072353             list.append(index)
%o A072353         x, y = y, x + y
%o A072353         index += 1
%o A072353     return list
%o A072353 print(A072353_list(57)) # _M. Eren Kesim_, Jul 19 2021
%Y A072353 Cf. A072351, A072352, A105564, A105566, A097348, A050815, A060384, A000045.
%K A072353 base,nonn
%O A072353 1,1
%A A072353 _Shyam Sunder Gupta_, Jul 17 2002
%E A072353 More terms from _Reinhard Zumkeller_, Apr 14 2005
%E A072353 Name edited by _Michel Marcus_, Jul 19 2021