OFFSET
1,1
COMMENTS
This is a sister sequence to A000057, because this sequence, since {k : A001177(k) = k-1}, might be called a subdiagonal sequence of A001177, and {k : A001177(k) = k+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
Are all members of this sequence prime? Using A069106, any composite members must exceed 89151931. - Robert Israel, Oct 13 2015
From Jianing Song, Jul 02 2019: (Start)
Yes, all terms are primes. See a brief proof below.
Also, if p == 1 (mod 4) then b(p) divides (p-Legendre(p,5))/2. So terms in this sequence are congruent to 11 or 19 modulo 20.
Primes p such that ord(-(3+sqrt(5))/2,p) = p-1, where ord(z,p) is the smallest integer k > 0 such that (z^k-1)/p is an algebraic integer. (End)
Comments from Amiram Eldar, Jan 30 2022 (Start)
Sequence A003147, "Primes p with a Fibonacci primitive root", is defined in the paper: Daniel Shanks, Fibonacci primitive roots, Fibonacci Quarterly, Vol. 10, No. 2 (1972), pp. 163-168, and 181.
A second paper on this subject Daniel Shanks and Larry Taylor, An Observation of Fibonacci Primitive Roots, Fibonacci Quarterly, Vol. 11, No. 2 (1973), pp. 159-160,
deals with terms p == 3 (mod 4) of A003147, i.e., the intersection of A003147 and A002145 (or A004767).
It states that if g is a Fibonacci primitive root of a prime p such that p == 3 (mod 4) then g-1 and g-2 are also primitive roots of p.
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.
Jianing Song, Proof that all terms are prime
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
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