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A001177
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Fibonacci entry points: a(n) = least k such that n divides Fibonacci number F_k (=A000045(k)).
(Formerly M2314 N0914)
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16
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1, 3, 4, 6, 5, 12, 8, 6, 12, 15, 10, 12, 7, 24, 20, 12, 9, 12, 18, 30, 8, 30, 24, 12, 25, 21, 36, 24, 14, 60, 30, 24, 20, 9, 40, 12, 19, 18, 28, 30, 20, 24, 44, 30, 60, 24, 16, 12, 56, 75, 36, 42, 27, 36, 10, 24, 36, 42, 58, 60, 15, 30, 24, 48, 35, 60, 68, 18, 24, 120
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| In the formula, the relation a(p^e) = p^(e-1)*a(p) is called Wall's conjecture, which has been verified for primes up to 10^14. See A060305. Primes for which this relation fails are called Wall-Sun-Sun primes. [From T. D. Noe (noe(AT)sspectra.com), Mar 03 2009]
Solutions to F_m == 0 (mod n) are given by m == 0 (mod a(n)).
If p is a prime of the form 10n +/- 1 then a(p) is a divisor of p-1. If q is a prime of the form 10n +/- 3 then a(q) is a divisor of q+1. - Robert G. Wilson v Jul 07 2007.
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REFERENCES
| A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 25.
Ramon Glez-Regueral, An entry-point algorithm for high-speed factorization, Thirteenth Internat. Conf. Fibonacci Numbers Applications, Patras, Greece, 2008.
B. H. Hannon and W. L. Morris, Tables of Arithmetical Functions Related to the Fibonacci Numbers. Report ORNL-4261, Oak Ridge National Laboratory, Oak Ridge, Tennessee, Jun 1968.
Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, Afterword by Herbert A. Hauptman, Nobel Laureate, 2. 'The Minor Modulus m(n)', Prometheus Books, NY, 2007, page 329-342.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
A. Allard, P. Lecomte, Periods and entry points in Fibonacci sequence, Fib. Quart. 17 (1) (1979) 51-57.
J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
B. H. Hannon and W. L. Morris, Tables of arithmetical functions related to the Fibonacci numbers, Math. Comp., 23 (1969), 459-460.
Eric Weisstein, MathWorld: Wall-Sun-Sun Prime [From T. D. Noe (noe(AT)sspectra.com), Mar 03 2009]
D. D. Wall, Fibonacci series modulo m, Am. Math. Monthly 67 (6) (1960) 525-532
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FORMULA
| A001175(n)=A001176(n)*a(n) for n >= 1.
a(n)=n iff n is of form 5^k or 12*5^k, a(n)=n-1 iff n is in A106535, a(n)=n+1 iff n is in A000057, a(n)=n+5 iff n is in 5*A000057, ... - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 10 2007
a(1)=1, a(2)=3, a(4)=6 and for e>2, a(2^e) = 3*2^(e-2); a(5^e) = 5^e; and if p is an odd prime not 5, then a(p^e) = p^max(0,e-s)*a(p) where s = valuation(A000045(a(p)),p) (Wall's conjecture states that s = 1 for all p). If (r,s)=1 then a(r*s)=LCM(a(r),a(s)). See Posamentier & Lahmann. - Robert G. Wilson v Jul 07 2007. Corrected by Max Alekseyev, Oct 19 2007, Jun 24 2011.
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MATHEMATICA
| f[n_] := Block[{k = 1}, While[ Mod[ Fibonacci@k, n] != 0, k++ ]; k]; Array[f, 74] (* Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 04 2007 *)
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PROG
| (PARI) a(n)=if(n<0, 0, s=1; while(fibonacci(s)%n>0, s++); s) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 10 2007
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CROSSREFS
| Cf. A000045, A060383, A001602. First occurrence of k is given in A131401.
Sequence in context: A016655 A057757 A058838 * A053991 A198617 A033957
Adjacent sequences: A001174 A001175 A001176 * A001178 A001179 A001180
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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