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A001583 Artiads: the primes p == 1 mod 5 for which Fibonacci((p-1)/5) is divisible by p.
(Formerly M5413 N2351)
18

%I M5413 N2351

%S 211,281,421,461,521,691,881,991,1031,1151,1511,1601,1871,1951,2221,

%T 2591,3001,3251,3571,3851,4021,4391,4441,4481,4621,4651,4691,4751,

%U 4871,5081,5281,5381,5531,5591,5641,5801,5881,6011,6101,6211,6271,6491,6841

%N Artiads: the primes p == 1 mod 5 for which Fibonacci((p-1)/5) is divisible by p.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D L. Tanner, Proc. London Math. Soc., 18 (1886-1887), 214-234; 24 (1892-1893), 223-262.

%H N. J. A. Sloane, <a href="/A001583/b001583.txt">Table of n, a(n) for n = 1..24903</a>, Apr 01, 2016 [First 1000 terms from T. D. Noe]

%H E. Lehmer, <a href="http://dx.doi.org/10.1016/0022-247X(66)90145-4">Artiads characterized</a>, J. Math. Anal. Appl. 15 1966 118-131.

%H E. Lehmer, <a href="/A001583/a001583b.pdf">Artiads characterized</a>, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

%H E. Lehmer, <a href="/A001583/a001583.pdf">On the quadratic character of the Fibonacci root</a>, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).

%p with(combinat): P:= proc(n) local p; p:=ithprime(n);

%p if p mod 5=1 and type(fibonacci((p-1)/5)/p,integer) then p; fi; end:

%p seq(P(i),i=1..10^4); # _Paolo P. Lava_, May 10 2017

%t Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* _Jean-Fran├žois Alcover_, Jun 22 2012 *)

%o (Haskell)

%o a001583 n = a001583_list !! (n-1)

%o a001583_list = filter

%o (\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list

%o -- _Reinhard Zumkeller_, Aug 15 2013

%o (PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]

%o list(lim)=my(v=List()); forprime(p=11,lim, if(p%5==1 && fibmod(p\5,p)==0, listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, Feb 06 2017

%Y Cf. A047650, A000045, A024894, subsequence of A030430.

%Y See also A270798 (a subsequence), A270800.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_.

%E More terms from _James A. Sellers_, Jan 25 2000

%E Edited by _N. J. A. Sloane_, Apr 01 2016

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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)