login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A038546 Numbers n such that n-th Fibonacci number has initial digits n. 5
0, 1, 5, 43, 48, 53, 3301, 48515, 348422, 406665, 1200207, 6698641, 190821326, 2292141445, 257125021372, 5843866639660, 45173327533483, 46312809996150, 59358981837795, 129408997210988, 1450344802530203, 5710154240910003 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The Mathematica coding used by Robert G. Wilson v implements Binet's Fibonacci number formula as suggested by David W. Wilson and incorporates Benoit Cloitre's use of logarithms to achieve a further increase in speed.

LINKS

Table of n, a(n) for n=1..22.

Ron Knott, Fibonacci Numbers and the Golden Section

Eric Weisstein's World of Mathematics, Fibonacci numbers

FORMULA

n>5 is in the sequence if a=(1+sqrt(5))/2 b=1/sqrt(5) and n==floor(b*(a^n)/10^(floor((log(b) +n*log(a))/log(10))-floor(log(n)/log(10))) ). - Benoit Cloitre, Feb 27 2002

EXAMPLE

a(3)=43 since 43rd Fibonacci number starts with 43 -> {43}3494437.

Fibonacci(53) is 53316291173, which begins with 53, so 53 is a term in the sequence.

MATHEMATICA

a = N[ Log[10, Sqrt[5]/5], 24]; b = N [Log[10, GoldenRatio], 24]; Do[ If[ IntegerPart[10^FractionalPart[a + n*b]*10^Floor[ Log[10, n]]] == n, Print[n]], {n, 225000000}] (* Robert G. Wilson v, May 09 2005 *)

(* confirmed with: *) fQ[n_] := (FromDigits[ Take[ IntegerDigits[ Fibonacci[n]], Floor[ Log[10, n] + 1]]] == n)

PROG

(PARI) /* To obtain terms > 5: */ a=(1+sqrt(5))/2; b=1/sqrt(5); for(n=1, 3500, if(n==floor(b*(a^n)/10^( floor(log(b *(a^n))/log(10))-floor(log(n)/log(10)))), print1(n, ", "))) \\ Benoit Cloitre, Feb 27 2002

CROSSREFS

Cf. A000045, A052000, A000350, A050816.

Sequence in context: A178614 A284170 A067927 * A022891 A106940 A106941

Adjacent sequences:  A038543 A038544 A038545 * A038547 A038548 A038549

KEYWORD

nonn,base,nice

AUTHOR

Jeff Burch

EXTENSIONS

Term a(6) from Patrick De Geest, Oct 15 1999

a(7) from Benoit Cloitre, Feb 27 2002

a(8)-a(11) from Robert G. Wilson v, May 09 2005

a(12) from Robert G. Wilson v, May 11 2005

More terms from Robert Gerbicz, Aug 22 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 20 04:23 EDT 2020. Contains 337264 sequences. (Running on oeis4.)