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A038546
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Numbers n such that n-th Fibonacci number has initial digits n.
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5
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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; internal format)
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OFFSET
| 1,3
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COMMENTS
| The Mathematica coding Robert G. Wilson v used is the Binet's Fibonacci number formula as suggested by David W. Wilson and further increase in speed by Benoit Cloitre's use of logarithms.
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LINKS
| R. Knott, Fibonacci Numbers and the Golden Section
Eric Weisstein's World of Mathematics, Fibonacci numbers
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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 (benoit7848c(AT)orange.fr), Feb 27 2002
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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.
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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}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), May 09 2005)
confirmed with fQ[n_] := (FromDigits[ Take[ IntegerDigits[ Fibonacci[n]], Floor[ Log[10, n] + 1]]] == n)
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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
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CROSSREFS
| Cf. A000045, A052000, A000350, A050816.
Sequence in context: A132487 A178614 A067927 * A022891 A106940 A106941
Adjacent sequences: A038543 A038544 A038545 * A038547 A038548 A038549
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KEYWORD
| nonn,base,nice
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AUTHOR
| Jeff Burch (gburch(AT)erols.com)
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EXTENSIONS
| Term a(6) from Patrick De Geest (pdg(AT)worldofnumbers.com), Oct 15 1999.
a(7) from Benoit Cloitre, Feb 27 2002
a(8), a(9), a(10) & a(11) from Robert G. Wilson v (rgwv(AT)rgwv.com), May 09 2005
a(12) from Robert G. Wilson v (rgwv(AT)rgwv.com), May 11 2005
More terms from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Aug 22 2006
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