A072352 -id:A072352 - cp's OEIS frontend

A072351 Smallest n-digit Fibonacci number.

1, 13, 144, 1597, 10946, 121393, 1346269, 14930352, 102334155, 1134903170, 12586269025, 139583862445, 1548008755920, 10610209857723, 117669030460994, 1304969544928657, 14472334024676221, 160500643816367088, 1100087778366101931, 12200160415121876738

Offset: 1

Shyam Sunder Gupta, Jul 17 2002

			a(3)=144, as 144 is smallest 3-digit Fibonacci number.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A072352, A105710, A105712, A105713, A105714, A105715, A105716, A105717, A105718, A105719.

F:= proc(n) option remember; local f;
      f:= `if`(n=1, [1$2], F(n-1));
      do f:= [f[2], f[1]+f[2]];
         if length(f[1]) `if`(n=1, 1, F(n-1)[2]):
seq(a(n), n=1..25);  # Alois P. Heinz, Mar 10 2016

a[n_] := Fibonacci[Ceiling[k /. FindRoot[Log[10, Fibonacci[k]] == n-1, {k, 1}]]]; Array[a, 20] (* Jean-François Alcover, Jan 18 2017 *)
With[{fbs=Fibonacci[Range[100]]},Table[SelectFirst[fbs,IntegerLength[#]==n&],{n,20}]] (* Harvey P. Dale, Dec 13 2024 *)

A072351(n,phi=(sqrt(5)+1)/2)=round(phi^ceil((n*log(10)+log(5)/2)/log(phi))/sqrt(5)) \\  Franklin T. Adams-Watters, May 27 2011

def A072351_list(n):
    list = [1]
    x, y = 1, 1
    while len(list) < n:
        if len(str(x)) < len(str(y)):
            list.append(y)
        x, y = y, x + y
    return list
print(A072351_list(20)) # M. Eren Kesim, Jun 28 2021

A072351(n) = floor(1/2 + phi^ceiling((n*log(10) + (1/2)*log(5))/log(phi))/sqrt(5)). - Franklin T. Adams-Watters, May 27 2011

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

Original entry on oeis.org

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

cp's OEIS Frontend

A072351 Smallest n-digit Fibonacci number.

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