A050815 -id:A050815 - cp's OEIS frontend

A060384 Number of decimal digits in n-th Fibonacci number.

1, 1, 1, 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, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Labos Elemer, Apr 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].
a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

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

A052005 Number of Fibonacci numbers (A000045) with length n in base 2.

Original entry on oeis.org

2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1

Offset: 1

Antti Karttunen and Patrick De Geest, Nov 15 1999

There are no double 2's except at the very start because multiplying by phi^3 adds at least 2 to Fn's binary length. For a similar reason there aren't any 3's because multiplying by phi^2 increments at least by one F(n)'s binary length.

Also a(n) is the number of Fibonacci numbers F(k) between powers of 2 such that 2^n <= F(k) < 2^(n+1). - Frank M Jackson, Apr 14 2013

			F(17) = 1597{10} = 11000111101{2} the only one of length 11 and F(18) = 2584{10} = 101000011000{2} the only one of length 12 so both a(11) and a(12) equal 1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000079, A001622, A052006, A000045, A050815, A036284, A037093, A022927, A022934, A104287.

nmax = 105; kmax = Floor[ k /. FindRoot[ Log[2, Fibonacci[k]] == nmax, {k, nmax, 2*nmax}]]; A052005 = Tally[ Length /@ IntegerDigits[ Fibonacci[ Range[kmax]], 2]][[All, 2]] (* Jean-François Alcover, May 07 2012 *)
termcount[n1_] := (m1=0; While[Fibonacci[m1]<2^n1, m1++]; m1); Table[termcount[n+1]-termcount[n], {n, 0, 200}] (* Frank M Jackson, Apr 14 2013 *)
Most[Transpose[Tally[Table[Length[IntegerDigits[Fibonacci[n], 2]], {n, 140}]]][[2]]] (* T. D. Noe, Apr 16 2013 *)

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(2)/log(phi) = A104287. - Amiram Eldar, Nov 21 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

A098842 Number of n-digit Fibonacci numbers.

Original entry on oeis.org

7, 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

Alex Vinokur (alexvn(AT)barak-online.net), Nov 03 2004

a(1)=7 because the Fibonacci numbers having one digit are the first seven; then the next five have two digits, so a(2)=5.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, The Fibonacci Numbers.
Ron Knott, [Number of digits in Fib(i)] : Calculator.

Essentially the same as A050815, which is the main entry for this sequence. Cf. A000045, A060384.

a098842 n = a098842_list !! (n-1)
a098842_list = map length $ group a060384_list
-- Reinhard Zumkeller, Mar 09 2013

Join[{7},Rest[Length[#]&/@Split[IntegerLength[Fibonacci[Range[510]]]]]] (* Harvey P. Dale, Jul 19 2020 *)

Corrected by Alexandre Wajnberg, Mar 28 2005

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

A052006 Numbers k for which Fibonacci(k) is the first member of a 1,1 pair (A052005).

Original entry on oeis.org

17, 30, 43, 53, 66, 79, 89, 102, 115, 125, 138, 151, 161, 174, 187, 200, 210, 223, 236, 246, 259, 272, 282, 295, 308, 321, 331, 344, 357, 367, 380, 393, 403, 416, 429, 442, 452, 465, 478, 488, 501, 514, 524, 537, 550, 560, 573, 586, 599, 609, 622, 635, 645

Offset: 0

Antti Karttunen and Patrick De Geest, Nov 15 1999

Keep adding the terms of sequence A052005 up to the first member of the next 1,1 pair to yield the terms of this sequence. - Patrick De Geest

Those k for which F(k-1) < 2^(floor(log_2(F(k)))) and F(k+1) >= 2^(floor(log_2(F(k)))+1) and F(k+2) >= 2^(floor(log_2(F(k)))+2).

Vincenzo Librandi, Table of n, a(n) for n = 0..1650

The first differences are A051392.

Cf. A052005, A000045, A050815, A036284, A037093.

With[{F = Fibonacci}, Reap[For[n=0, n<1000, n++, If[F[n-1] < 2^Floor[Log[2, F[n]]] && F[n+1] >= 2^(Floor[Log[2, F[n]]]+1) && F[n+2] >= 2^(Floor[Log[ 2, F[n]]]+2), Print[n]; Sow[n]]]][[2, 1]]] (* Jean-François Alcover, Feb 27 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.

A117768 Number of Lucas numbers with n digits.

Original entry on oeis.org

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: 0

Luc Stevens (lms022(AT)yahoo.com), Apr 15 2006

Cf. A000204. Same as A050815 and A098842 but without the first 3 numbers.

cp's OEIS Frontend

A060384 Number of decimal digits in n-th Fibonacci number.

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

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A052005 Number of Fibonacci numbers (A000045) with length n in base 2.

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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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A098842 Number of n-digit Fibonacci numbers.

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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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A052006 Numbers k for which Fibonacci(k) is the first member of a 1,1 pair (A052005).

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