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A106535 Numbers n such that the smallest x > 1 for which Fibonacci(x) = 0 mod n is x = n - 1. 7
11, 19, 31, 59, 71, 79, 131, 179, 191, 239, 251, 271, 311, 359, 379, 419, 431, 439, 479, 491, 499, 571, 599, 631, 659, 719, 739, 751, 839, 971, 1019, 1039, 1051, 1091, 1171, 1259, 1319, 1399, 1439, 1451, 1459, 1499, 1531, 1559, 1571, 1619, 1759, 1811, 1831 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is a sister sequence to A000057, because this sequence, since A001177(n) = n-1, might be called a subdiagonal sequence of A001177, and A001177(n) = n+1, which might be called a superdiagonal sequence of A001177. Sequences A000057 and A106535 are disjoint. Is this sequence the set of all divisors of some family of sequences, like A000057 is? - Art DuPre, Jul 11 2012

Subset of A219612. - Paolo P. Lava, Dec 22 2014

Are all members of this sequence prime?  Using A069106, any composite members must exceed 89151931. - Robert Israel, Oct 13 2015

LINKS

Robert Israel, Table of n, a(n) for n = 1..2000

Alfred Brousseau, Primes which are factors of all Fibonacci sequences, Fib. Quart., 2 (1964), 33-38.

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972.

FORMULA

{n: A001177(n) = n-1}. - R. J. Mathar, Jul 09 2012

MAPLE

A106535 := proc(n)

        option remember;

        if n = 1 then

                11;

        else

                for a from procname(n-1)+1 do

                        if A001177(a) = a-1 then

                                return a;

                        end if;

                end do:

        end if;

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

# Alternative:

fmod:= proc(a, b) local A;

  uses LinearAlgebra[Modular];

  A:= Mod(b, <<1, 1>|<1, 0>>, integer[8]);

  MatrixPower(b, A, a)[1, 2];

end proc:

filter:= proc(n)

  local cands;

  if fmod(n-1, n) <> 0 then return false fi;

  cands:= map(t -> (n-1)/t, numtheory:-factorset(n-1));

  andmap(c -> (fmod(c, n) > 0), cands);

end proc:

select(filter, [$2..10^4]); # Robert Israel, Oct 13 2015

MATHEMATICA

f[n_] := Block[{x = 2}, While[Mod[Fibonacci@ x, n] != 0, x++]; x]; Select[Range@ 1860, f@ # == # - 1 &] (* Michael De Vlieger, Oct 13 2015 *)

PROG

(GAP) Filtered([2..2000], n -> Fibonacci(n-1) mod n = 0 and Filtered( [2..n-2], x -> Fibonacci(x) mod n = 0 ) = [] );

(PARI) isok(n) = {x = 2; while(fibonacci(x) % n, x++); x == n-1; } \\ Michel Marcus, Oct 20 2015

CROSSREFS

Cf. A000057, A001175, A069106.

Sequence in context: A152091 A272550 A122869 * A178150 A214784 A205798

Adjacent sequences:  A106532 A106533 A106534 * A106536 A106537 A106538

KEYWORD

nonn

AUTHOR

Peter K. Pearson (ppearson+att(AT)spamcop.net), May 06 2005

EXTENSIONS

Corrected by T. D. Noe, Oct 25 2006

STATUS

approved

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Last modified September 25 12:56 EDT 2017. Contains 292469 sequences.