1,2

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]

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 25.

J. D. Fulton and W. L. Morris, On arithmetical functions related to the Fibonacci numbers, Acta Arithmetica, 16 (1969), 105-110.

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.

Review of B. H. Hannon and W. L. Morris tables, Math. Comp., 23 (1969), 459-460.

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.

T. D. Noe, Table of n, a(n) for n=1..10000

Eric Weisstein, MathWorld: Wall-Sun-Sun Prime [From T. D. Noe (noe(AT)sspectra.com), Mar 03 2009]

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^(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.

f[n_] := Block[{k = 1}, While[ Mod[ Fibonacci@k, n] != 0, k++ ]; k]; Array[f, 74] (* Robert G. Wilson v (rgwv@rgwv.com), Jul 04 2007 *)

(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

Cf. A000045, A060383, A001602. First occurrence of k is given in A131401.

Adjacent sequences: A001174 A001175 A001176 this_sequence A001178 A001179 A001180

Sequence in context: A016655 A057757 A058838 this_sequence A053991 A033957 A031131

nonn

N. J. A. Sloane (njas(AT)research.att.com).