%I A001177 M2314 N0914
%S A001177 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,
%T A001177 21,36,24,14,60,30,24,20,9,40,12,19,18,28,30,20,24,44,30,60,24,16,12,
%U A001177 56,75,36,42,27,36,10,24,36,42,58,60,15,30,24,48,35,60,68,18,24,120
%N A001177 Fibonacci entry points: a(n) = least k such that n divides Fibonacci 
               number F_k.
%C A001177 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]
%D A001177 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001177 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001177 A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci 
               Association, San Jose, CA, 1972, p. 25.
%D A001177 J. D. Fulton and W. L. Morris, On arithmetical functions related to the 
               Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.
%D A001177 Ramon Glez-Regueral, An entry-point algorithm for high-speed factorization, 
               Thirteenth Internat. Conf. Fibonacci Numbers Applications, Patras, 
               Greece, 2008.
%D A001177 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.
%D A001177 Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 
               459-460.
%D A001177 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.
%H A001177 T. D. Noe, <a href="b001177.txt">Table of n, a(n) for n=1..10000</a>
%H A001177 Eric Weisstein, <a href="http://mathworld.wolfram.com/Wall-Sun-SunPrime.html">
               MathWorld: Wall-Sun-Sun Prime</a> [From T. D. Noe (noe(AT)sspectra.com), 
               Mar 03 2009]
%F A001177 A001175(n)=A001176(n)*a(n) for n >= 1.
%F A001177 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
%F A001177 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^(e-1)*a(p). 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. 
               If (r,s)=1 then a(r*s)=LCM(a(r),a(s)). Solutions to F_k == 0 (mod 
               n) are given by k == 0 (mod a(n)). See Posamentier & Lahmann. - Robert 
               G. Wilson v Jul 07 2007. Corrected by Max Alekseyev, Oct 19 2007.
%t A001177 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 
               *)
%o A001177 (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
%Y A001177 Cf. A000045, A060383, A001602. First occurrence of k is given in A131401.
%Y A001177 Sequence in context: A016655 A057757 A058838 this_sequence A053991 A033957 
               A031131
%Y A001177 Adjacent sequences: A001174 A001175 A001176 this_sequence A001178 A001179 
               A001180
%K A001177 nonn
%O A001177 1,2
%A A001177 N. J. A. Sloane (njas(AT)research.att.com).