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A106535
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Numbers k such that the smallest x > 1 for which Fibonacci(x) == 0 mod k is x = k - 1.
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16
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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
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OFFSET
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1,1
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COMMENTS
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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
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)
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,
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.
The first 2000 terms of (A003147 intersect A002145) agree with the present sequence, although the definitions are quite different. Are these two sequences the same? (End)
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LINKS
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FORMULA
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MAPLE
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option remember;
if n = 1 then
11;
else
for a from procname(n-1)+1 do
return a;
end if;
end do:
end if;
# 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:
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MATHEMATICA
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f[n_] := Block[{x = 2}, While[Mod[Fibonacci@ x, n] != 0, x++]; x]; Select[Range@ 1860, f@ # == # - 1 &] (* Michael De Vlieger, Oct 13 2015 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Peter K. Pearson (ppearson+att(AT)spamcop.net), May 06 2005
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EXTENSIONS
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STATUS
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approved
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