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%I #25 Jul 08 2022 11:06:45
%S 0,1,5,43,48,53,3301,48515,348422,406665,1200207,6698641,190821326,
%T 2292141445,257125021372,5843866639660,45173327533483,46312809996150,
%U 59358981837795,129408997210988,1450344802530203,5710154240910003
%N Numbers n such that n-th Fibonacci number has initial digits n.
%C 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.
%C Fixed points of A020344. - _Alois P. Heinz_, Jul 08 2022
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fib.html">Fibonacci Numbers and the Golden Section</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci numbers</a>
%F 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
%e a(3)=43 since 43rd Fibonacci number starts with 43 -> {43}3494437.
%e Fibonacci(53) is 53316291173, which begins with 53, so 53 is a term in the sequence.
%t 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 *)
%t (* confirmed with: *) fQ[n_] := (FromDigits[ Take[ IntegerDigits[ Fibonacci[n]], Floor[ Log[10, n] + 1]]] == n)
%o (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
%Y Cf. A000045, A020344, A052000, A000350, A050816.
%K nonn,base,nice
%O 1,3
%A _Jeff Burch_
%E Term a(6) from _Patrick De Geest_, Oct 15 1999
%E a(7) from _Benoit Cloitre_, Feb 27 2002
%E a(8)-a(11) from _Robert G. Wilson v_, May 09 2005
%E a(12) from _Robert G. Wilson v_, May 11 2005
%E More terms from _Robert Gerbicz_, Aug 22 2006
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