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 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 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 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

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