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A001177 Fibonacci entry points: a(n) = least k such that n divides Fibonacci number F_k (=A000045(k) for k >= 1).
(Formerly M2314 N0914)
41
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; text; internal format)
OFFSET

1,2

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. - T. D. Noe, Mar 03 2009

All solutions to F_m == 0 (mod n) are given by m == 0 (mod a(n)). For a proof see, e.g., Vajda, p. 73. [Old comment changed by Wolfdieter Lang, Jan 19 2015]

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

Definition 1 in Riasat (2011) calls this k(n), or sometimes just k. Corollary 1 in the same paper, "every positive integer divides infinitely many Fibonacci numbers," demonstrates that this sequence is infinite. - Alonso del Arte, Jul 27 2013

If p is a prime then a(p)<=p+1. This is because if p is a prime then exactly one of the following Fibonacci numbers is a multiple of p: F(p-1), F(p) or F(p+1). - Dmitry Kamenetsky, Jul 23 2015

From Renault 1996:

  1. a(gcd(n,m))=gcd(a(n),a(m)).

  2. if n|m then a(n)|a(m).

  3. if m has prime factorization m=p1^e1 * p2^e2 * ... * pn^en then a(m)=lcm(a(p1^e1), a(p2^e2), ..., a(pn^en)). - Dmitry Kamenetsky, Jul 23 2015

a(n)=n if and only if n=5^k or n=12*5^k for some k>=0 (see Marques 2012). - Dmitry Kamenetsky, Aug 08 2015

Every positive integer (except 2) eventually appears in this sequence. This is because every Fibonacci number bigger than 1 (except F(6)=8 and F(12)=144) has at least one prime factor that is not a factor of any earlier Fibonacci number (see Knott reference). Let f(n) be such a prime factor for F(n); then a(f(n))=n. - Dmitry Kamenetsky, Aug 08 2015

We can reconstruct the Fibonacci numbers from this sequence using the formula F(n+2) = 1 + Sum_{i: a(i) <= n} phi(i)*floor(n/a(i)), where phi(n) is Euler's totient function A000010 (see the Stroinski link). For example F(6) = 1 + phi(1)*floor(4/a(1)) + phi(2)*floor(4/a(2)) + phi(3)*floor(4/a(4)) = 1 + 1*4 + 1*1 + 2*1 = 8. - Peter Bala, Sep 10 2015

REFERENCES

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

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, June 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).

S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

N. N. Vorob'ev, Fibonacci numbers, Blaisdell, NY, 1961.

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.

R. C. Archibald (?), Review of B. H. Hannon and W. L. Morris, Tables of arithmetical functions related to the Fibonacci numbers, Math. Comp., 23 (1969), 459-460.

B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014

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 [Annotated and scanned copy]

Ron Knott, The first 300 Fibonacci numbers factorized

Diego Marques, Fixed points of the order of appearance in the Fibonacci sequence, Fibonacci Quart. 50:4 (2012), pp. 346-352.

Diego Marques, The order of appearance of the product of consecutive Lucas numbers, Fibonacci Quarterly, 51 (2013), 38-43.

Renault, The Fibonacci sequence under various moduli, Masters Thesis, Wake Forest University, 1996.

Samin Riasat, Z[phi] and the Fibonacci Sequence Modulo n, Mathematical Reflections 1 (2011): 1 - 7.

U. Stroinski, Connection between Euler's totient function and Fibonacci numbers, Mathematics Stack Exchange, Feb 17 2015

D. D. Wall, Fibonacci series modulo m, Am. Math. Monthly 67 (6) (1960) 525-532

Eric Weisstein, MathWorld: Wall-Sun-Sun Prime

FORMULA

A001175(n) = A001176(n) * a(n) for n >= 1.

a(n) = n if and only if n is of form 5^k or 12*5^k (proved in Marques paper), a(n) = n - 1 if and only if n is in A106535, a(n) = n + 1 if and only if n is in A000057, a(n) = n + 5 if and only if n is in 5*A000057, ... - Benoit Cloitre, 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 (m, n) = 1 then a(m*n) = LCM(a(m), a(n)). See Posamentier & Lahmann. - Robert G. Wilson v, Jul 07 2007; corrected by Max Alekseyev, Oct 19 2007, Jun 24 2011

Apparently a(n) = A213648(n) + 1 for n >= 2. - Art DuPre, Jul 01 2012

a(n)<n^2. [Vorob'ev] - Zak Seidov, Jan 07 2016

EXAMPLE

a(4) = 6 because the smallest Fibonacci number that 4 divides is F(6) = 8.

a(5) = 5 because the smallest Fibonacci number that 5 divides is F(5) = 5.

a(6) = 12 because the smallest Fibonacci number that 6 divides is F(12) = 144.

From Wolfdieter Lang, Jan 19 2015: (Start)

a(2) = 3, hence 2 | F(m) iff m = 2*k, for k >= 0;

a(3) = 4, hence 3 | F(m) iff m = 4*k, for k >= 0;

etc. See a comment above with the Vajda reference.

(End)

MAPLE

A001177 := proc(n)

        for k from 1 do

                if combinat[fibonacci](k) mod n = 0 then

                        return k;

                end if;

        end do:

end proc: # R. J. Mathar, Jul 09 2012

N:= 1000: # to get a(1) to a(N)

L:= ilcm($1..N):

count:= 0:

for n from 1 while count < N do

  fn:= igcd(L, combinat:-fibonacci(n));

  divs:= select(`<=`, numtheory:-divisors(fn), N);

  for d in divs do if not assigned(A[d]) then count:= count+1; A[d]:= n fi od:

od:

seq(A[n], n=1..N); # Robert Israel, Oct 14 2015

MATHEMATICA

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

PROG

(PARI) a(n)=if(n<0, 0, s=1; while(fibonacci(s)%n>0, s++); s) \\ Benoit Cloitre, Feb 10 2007

(PARI) ap(p)=my(k=1); while(fibonacci(k++)%p, ); k

a(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, ap(f[i, 1]^f[i, 2]), ap(f[i, 1])*f[i, 1]^(f[i, 2]-1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<<max(f[1, 2]-2, 1)); lcm(v) \\ Charles R Greathouse IV, Feb 04 2014

(PARI) ap(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, c+o]; k++); k

a(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, ap(f[i, 1]^f[i, 2]), ap(f[i, 1])*f[i, 1]^(f[i, 2]-1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<<max(f[1, 2]-2, 1)); lcm(v) \\ Charles R Greathouse IV, Feb 13 2014

(Scheme) (define (A001177 n) (let loop ((k 1)) (cond ((zero? (modulo (A000045 k) n)) k) (else (loop (+ k 1)))))) ;; Antti Karttunen, Dec 21 2013

(Haskell)

a001177 n = head [k | k <- [1..], a000045 k `mod` n == 0]

-- Reinhard Zumkeller, Jan 15 2014

CROSSREFS

Cf. A000045, A001175, A001176, A060383, A001602. First occurrence of k is given in A131401. A233281 gives such k that a(k) is a prime.

From Antti Karttunen, Dec 21 2013: (Start)

Various derived sequences:

A047930(n) = A000045(a(n)).

A037943(n) = A000045(a(n))/n.

A217036(n) = A000045(a(n)-1) mod n.

A132632(n) = a(n^2).

A132633(n) = a(n^3).

A214528(n) = a(n!).

A215011(n) = a(A000217(n)).

A215453(n) = a(n^n).

Analogous sequence for the tribonacci numbers: A046737, for Lucas numbers: A223486, for Pell numbers: A214028.

Cf. also A000057, A106535, A120255, A120256, A175026, A213648, A214031, A214781, A214783, A230359, A233283, A233285, A233287. (End)

Sequence in context: A016655 A057757 A058838 * A053991 A276814 A198617

Adjacent sequences:  A001174 A001175 A001176 * A001178 A001179 A001180

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Added 'for k >= 1' in the name (for k >= 0, a(0) = 0). - Wolfdieter Lang, Jan 19 2015

STATUS

approved

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Last modified December 4 17:40 EST 2016. Contains 278755 sequences.