A038546 Numbers n such that n-th Fibonacci number has initial digits n.

0, 1, 5, 43, 48, 53, 3301, 48515, 348422, 406665, 1200207, 6698641, 190821326, 2292141445, 257125021372, 5843866639660, 45173327533483, 46312809996150, 59358981837795, 129408997210988, 1450344802530203, 5710154240910003

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.

Fixed points of A020344. - Alois P. Heinz, Jul 08 2022

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, A020344, A052000, A000350, A050816.

Sequence in context: A360942 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