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A217036 Term preceding the first zero in the Fibonacci numbers modulo n. 4
1, 2, 1, 3, 5, 6, 5, 8, 7, 1, 5, 8, 13, 11, 9, 4, 17, 1, 9, 13, 1, 22, 17, 18, 5, 26, 13, 1, 11, 1, 17, 23, 21, 6, 17, 31, 1, 14, 29, 40, 13, 42, 1, 26, 45, 46, 41, 48, 7, 35, 25, 23, 53, 34, 41, 20, 1, 1, 41, 11, 1, 55, 33, 47, 23, 66, 33, 22, 41, 1, 17, 27 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

The multiplicative order of term n modulo n is given by sequence A001176.

Let M = [{1, 1}, {1, 0}], I = [{1, 0}, {0, 1}] is the 2 X 2 identity matrix, then A001177(n) is the smallest k > 0 such that M^k == r*I (mod n) for some r such that 0 <= r < n, and a(n) gives the value r. - Jianing Song, Jul 04 2019

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 2..10000

FORMULA

a(n) = F(G(n)-1) mod n where G(n) is sequence A001177 and F(m) is the m-th Fibonacci number. In particular, if n is a Fibonacci number, the n-th term is the previous Fibonacci number.

From Jianing Song, Jul 04 2019:

Also a(n) = F(G(n)+1) mod n.

a(2^e) = 1 if e = 1, 2, 2^(e-1) + 1 if e >= 3; a(p^e) = a(p)^(p^(e-1)) mod p^e for odd primes p.

For odd primes p, a(p^e) = 1 if and only if A001177(p) == 2 (mod 4); a(p^e) = p^e - 1 if and only if 4 divides A001177(p). (End)

MAPLE

a:= proc(n) local f, g; f, g:= 1, 0;

      while f<>0 do f, g:= irem(f+g, n), f od; g

    end:

seq(a(n), n=2..100);  # Alois P. Heinz, Sep 24 2012

MATHEMATICA

Table[k = 1; While[Mod[Fibonacci[k], n] > 0, k++]; Mod[Fibonacci[k - 1], n], {n, 2, 100}] (* T. D. Noe, Sep 24 2012 *)

PROG

(PARI) a(n)=my(a=0, b=1); for(k=1, n^2, [a, b]=[b, (a+b)%n]; if(!b, return(a))) \\ Charles R Greathouse IV, Sep 24 2012

CROSSREFS

Sequence in context: A327452 A257982 A275705 * A127201 A225844 A006769

Adjacent sequences:  A217033 A217034 A217035 * A217037 A217038 A217039

KEYWORD

nonn

AUTHOR

David Spies, Sep 24 2012

EXTENSIONS

a(14)-a(70) from Charles R Greathouse IV, Sep 24 2012

STATUS

approved

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Last modified May 31 04:43 EDT 2020. Contains 334747 sequences. (Running on oeis4.)