A105566 -id:A105566 - cp's OEIS frontend

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

6, 11, 16, 20, 25, 30, 35, 39, 44, 49, 54, 59, 63, 68, 73, 78, 83, 87, 92, 97, 102, 106, 111, 116, 121, 126, 130, 135, 140, 145, 150, 154, 159, 164, 169, 173, 178, 183, 188, 193, 197, 202, 207, 212, 216, 221, 226, 231, 236, 240, 245, 250, 255, 260, 264, 269, 274

Offset: 1

Shyam Sunder Gupta, Jul 17 2002

Partial sums of A050815: a(n) = Sum_{k=1..n} A050815(k). - Reinhard Zumkeller, Apr 14 2005

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

			a(3)=16, as the 16th Fibonacci number is the largest Fibonacci number with 3 digits.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhäuser 2003).
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A072351, A072352, A105564, A105566, A097348, A050815, A060384, A000045.

With[{fibs=Fibonacci[Range[300]]},Flatten[Position[fibs,#]&/@ Table[ Max[ Select[fibs,IntegerLength[#]==n&]],{n,60}]]] (* Harvey P. Dale, Nov 09 2011 *)

def A072353_list(n):
    list = []
    x, y, index = 1, 1, 1
    while len(list) < n:
        if len(str(x)) < len(str(y)):
            list.append(index)
        x, y = y, x + y
        index += 1
    return list
print(A072353_list(57)) # M. Eren Kesim, Jul 19 2021

Limit_{n->oo} a(n)/n = 1/log_10((1+sqrt(5))/2) = 1/A097348 = 4.784... . - Reinhard Zumkeller, Apr 14 2005.

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

More terms from Reinhard Zumkeller, Apr 14 2005

Name edited by Michel Marcus, Jul 19 2021

A105565 a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 14 2005

The sequence is almost periodic, see also A105566;
a(n) = 1 - A105563(n), for n>1.
a(n) = A050815(n) - 4, for n>1. - Hans J. H. Tuenter, Jul 29 2025

			The Fibonacci numbers with two decimal digits are 13, 21, 34, 55, 89; a total of five, so that a(2)=1.

Robert Israel, Table of n, a(n) for n = 1..10000
Igor Szczyrba, Rafał Szczyrba, and Martin Burtscher, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhäuser 2003).
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Almost Periodic Function

Cf. A050815, A060384, A000045.

n:= 1: count:= 2: a:= 0: b:= 1:
for m from 2 while n < 101 do
  c:= b; b:= a+b; a:= c;
  s:= ilog10(b)+1;
  if s = n then count:= count+1
  else
    if count = 5 then A[n]:= 1 else A[n]:= 0 fi;
    count:= 1; n:= s
  fi
od:
seq(A[i],i=1..100); # Robert Israel, Dec 17 2018

For n>1, a(n) = [{n*alpha+beta}<{alpha}], where alpha=log(10)/log(phi), beta=log(5)/(2*log(phi)), [X] is the Iverson bracket, {x}=x-floor(x), denotes the fractional part of x, and phi=(1+sqrt(5))/2. - Hans J. H. Tuenter, Jul 29 2025

A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18

Offset: 1

Reinhard Zumkeller, Apr 14 2005

nonn

Lim_{n->infinity} a(n)/n = 5 - 1/log_10((1+sqrt(5))/2) = 0.215....

a(n) = Sum_{k=1..n} A105563(k); a(n) = n - A105566(n).

Juergen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhäuser 2003).

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

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A105565 a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

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A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.

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