A105564 -id:A105564 - 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

A105563 a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 14 2005

The sequence is almost periodic, see also A105564;

Asymptotically, a fraction of 1-alpha=0.215028... of the terms are 1. For the partial sums S(n) = Sum_{k=1..n} a(k), this implies S(n)~(1-alpha)*n. Conjecture: -beta < S(n)-(1-alpha)*n < 1-beta. The constants alpha and beta are as defined in the formula section. - Hans J. H. Tuenter, Aug 28 2025

Hans J. H. Tuenter, Table of n, a(n) for n = 1..10000
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik, 58(1):26-33, 2003.
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Almost Periodic Function

Cf. A000045, A050815, A060384, A105565, A001622.

If[#==4,1,0]&/@Tally[IntegerLength/@Fibonacci[Range[500]]][[;;,2]] (* Harvey P. Dale, Nov 15 2023 *)

a(n) = 1 - A105565(n), for n>1.
a(n) = 5 - A050815(n), for n>1. - Hans J. H. Tuenter, Aug 28 2025
For n>1, a(n) = [{n*alpha+beta}>alpha], where alpha=log(10)/log(phi)-4, beta=log(5)/(2*log(phi))-1, [] is the Iverson bracket, {x}=x-floor(x), denotes the fractional part of x, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 28 2025

A105566 Number of blocks of exactly 5 Fibonacci numbers having equal length <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 49, 50, 51, 52, 53, 53, 54, 55, 56, 56, 57

Offset: 1

Reinhard Zumkeller, Apr 14 2005

nonn

a(n) = Sum_{k=1..n} A105565(k); a(n) = n - A105564(n);

lim_{n->inf} a(n)/n = 1/log_10((1+sqrt(5))/2) - 4 = 0.784....

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

Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A105564, A072353, A050815, A060384, A000045.

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

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A105563 a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

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A105566 Number of blocks of exactly 5 Fibonacci numbers having equal length <= n.

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