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 A106535 Numbers k such that the smallest x > 1 for which Fibonacci(x) = 0 mod k is x = k - 1. 15
 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 {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 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 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) 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 Cf. A000057, A001175, A069106. Similar sequences that give primes p such that A001177(p) = (p-1)/s: this sequence (s=1), A308795 (s=2), A308796 (s=3), A308797 (s=4), A308798 (s=5), A308799 (s=6), A308800 (s=7),A308801 (s=8), A308802 (s=9). 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 October 23 05:56 EDT 2019. Contains 328335 sequences. (Running on oeis4.)