A072353 -id:A072353 - cp's OEIS frontend

A072354 a(n)-th Fibonacci number is the smallest Fibonacci number containing n digits.

1, 7, 12, 17, 21, 26, 31, 36, 40, 45, 50, 55, 60, 64, 69, 74, 79, 84, 88, 93, 98, 103, 107, 112, 117, 122, 127, 131, 136, 141, 146, 151, 155, 160, 165, 170, 174, 179, 184, 189, 194, 198, 203, 208, 213, 217, 222, 227, 232, 237

Offset: 1

Shyam Sunder Gupta, Jul 18 2002

			a(3) = 12 as the 12th Fibonacci number is the smallest Fibonacci number with 3 digits.

Harry J. Smith, Table of n, a(n) for n = 1..20899

Cf. A001622, A105700, A105702, A105703, A105704, A105705, A105706, A105707, A105708, A105709.

Flatten[Table[Position[IntegerLength[Fibonacci[Range[250]]],n,{1},1],{n,50}]] (* Harvey P. Dale, Dec 22 2015 *)

a(n)={my(k=1); while(logint(fibonacci(k),10)Harry J. Smith, Nov 29 2008

For n>1, a(n) = A072353(n-1) + 1. - Michel Marcus, Jun 01 2014

For n>1, a(n) = ceiling(n*log(10)/log(phi)-log(20)/(2*log(phi))), where phi=(1+sqrt(5))/2, the golden ratio. - Hans J. H. Tuenter, Jul 13 2025

A050815 Number of positive Fibonacci numbers with n decimal digits.

Original entry on oeis.org

6, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5

Offset: 1

Patrick De Geest, Oct 15 1999

If n>1 then a(n) = 4 or 5. - Robert Gerbicz, Sep 05 2002

The sequence is almost periodic, see also A072353. - Reinhard Zumkeller, Apr 14 2005

			At length 1 there are 6 such numbers: 1, 1, 2, 3, 5 and 8.

Hans J. H. Tuenter, Table of n, a(n) for n = 1..1001
Andreas Guthmann, Wieviele k-stellige Fibonaccizahlen gibt es?, Archiv der Mathematik, Vol. 59, No. 4 (1992), pp. 334-340.
Ron Knott, The Fibonacci Numbers.
Ron Knott, Fibonacci Numbers and the Golden Section.
Ron Knott, [Number of digits in Fib(i)] : Calculator.
Jan-Christoph Puchta, The Number of k-Digit Fibonacci Numbers, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), pp. 334-335.
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik, Vol. 58 (Birkhäuser 2003), pp. 26-33.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Almost Periodic Function.

See A098842 for another version.

Cf. A000045, A001622, A002390, A072354, A105563, A105565, A060384, A097348.

Drop[Last/@Tally[Table[IntegerLength[Fibonacci[n]],{n,505}]],-1] (* Jayanta Basu, Jun 01 2013 *)

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = log(10)/log(phi) = 1/A097348 = 4.7849719667... - Amiram Eldar, Jan 12 2022
For n>1, a(n) = 4+[{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 20 2025
a(n) = A072354(n+1)-A072354(n), a first-order difference. - Hans J. H. Tuenter, Jul 20 2025

A072352 a(n) is the largest n-digit Fibonacci number.

Original entry on oeis.org

8, 89, 987, 6765, 75025, 832040, 9227465, 63245986, 701408733, 7778742049, 86267571272, 956722026041, 6557470319842, 72723460248141, 806515533049393, 8944394323791464, 99194853094755497, 679891637638612258, 7540113804746346429, 83621143489848422977

Offset: 1

Shyam Sunder Gupta, Jul 17 2002

			a(3)=987, as 987 is largest 3-digit Fibonacci number.

Robert Israel, Table of n, a(n) for n = 1..990

Cf. A000045, A072351, A072353.

fib:= combinat:-fibonacci:
g:= proc(x) local n;
  n:= floor(ln((2*x+1)*sqrt(5)/2)/ln((1+sqrt(5))/2));
  if fib(n) > x then while fib(n) > x do n:= n-1 od
  elif fib(n+1) <= x then while fib(n+1) <= x do n:= n+1 od
  fi;
  fib(n)
end:
seq(g(10^n),n=1..50); # Robert Israel, Mar 10 2016
# second Maple program:
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]) F(n)[1]:
seq(a(n), n=1..25);  # Alois P. Heinz, Mar 10 2016

Table[k=1;While[Fibonacci@++k<10^n];Fibonacci[k-1],{n,20}] (* Giorgos Kalogeropoulos, Jul 06 2021 *)

def A072352_list(n):
    list = []
    x, y = 1, 1
    while len(list) < n:
        if len(str(x)) < len(str(y)):
            list.append(x)
        x, y = y, x + y
    return list
print(A072352_list(20)) # M. Eren Kesim, Jun 28 2021

a(n) = A000045(A072353(n)). - Robert Israel, Mar 10 2016

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.

A072509 Number of Fibonacci numbers F(k) <= 10^n which end in 1.

Original entry on oeis.org

2, 2, 3, 3, 4, 5, 6, 6, 6, 7, 7, 7, 8, 10, 11, 11, 11, 13, 13, 14, 14, 15, 15, 15, 15, 17, 18, 19, 19, 20, 21, 22, 22, 22, 23, 23, 23, 23, 26, 27, 27, 27, 29, 29, 30, 30, 31, 31, 31, 31, 32, 34, 35, 35, 36, 37, 38, 38, 38, 39, 39, 39, 39, 42, 42, 43, 43, 45, 45, 46, 46, 47, 47

Offset: 0

Vladeta Jovovic, Aug 23 2002

Note that F(k) ends in 1 if and only if k == 1, 2, 8, 19, 22, 28, 41, or 59 (mod 60). - Robert Israel, May 14 2018

Robert Israel, Table of n, a(n) for n = 0..10000

Different from A073550. Cf. A072353, A072675.

N:= 100: m:= 0:
A:= Array(0..N):
A[0]:= 2:
for i from 0 while m <= N do
  for j in [1, 2, 8, 19, 22, 28, 41, 59] do
    m:= ilog10(combinat:-fibonacci(60*i+j))+1;
    if m > N then break fi;
    A[m..N]:= A[m..N]+1;
od od:
convert(A,list); # Robert Israel, May 14 2018

With[{s = Array[Fibonacci, 350]}, Table[Count[TakeWhile[s, # <= 10^n &], ?(Mod[#, 10] == 1 &)], {n, 0, IntegerLength@ Max@ s}] ] (* _Michael De Vlieger, May 14 2018 *)

A164018 The index values of the smallest and the largest n-digit Fibonacci numbers.

Original entry on oeis.org

0, 6, 7, 11, 12, 16, 17, 20, 21, 25, 26, 30, 31, 35, 36, 39, 40, 44, 45, 49, 50, 54, 55, 59, 60, 63, 64, 68, 69, 73, 74, 78, 79, 83, 84, 87, 88, 92, 93, 97, 98, 102, 103, 106, 107, 111, 112, 116, 117, 121, 122, 126, 127, 130, 131, 135, 136, 140, 141, 145

Offset: 0

Parthasarathy Nambi, Aug 07 2009

= A072354 + A072353 [From Parthasarathy Nambi, Sep 14 2009]

			The index value of the smallest ten digit Fibonacci number is 45. The index value of the largest ten digit Fibonacci number is 49.

Harvey P. Dale, Table of n, a(n) for n = 0..999

Cf. A000045

Drop[Flatten[{#-1,#}&/@Flatten[Table[Position[IntegerLength[Fibonacci[ Range[250]]],n,1,1],{n,50}]]],{2}](* Harvey P. Dale, Jun 04 2018 *)

More terms from Harvey P. Dale, Jun 04 2018

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A072354 a(n)-th Fibonacci number is the smallest Fibonacci number containing n digits.

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A050815 Number of positive Fibonacci numbers with n decimal digits.

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A072352 a(n) is the largest n-digit Fibonacci number.

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

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A072509 Number of Fibonacci numbers F(k) <= 10^n which end in 1.

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A164018 The index values of the smallest and the largest n-digit Fibonacci numbers.

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