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%I M5413 N2351 #110 Feb 13 2024 06:54:14
%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.
%C From _A.H.M. Smeets_, Nov 15 2023: (Start)
%C Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
%C In E. Lehmer, Artiads characterized, she counted in the table on p. 122 the primes p for which p == 1 (mod 5) instead of all primes. As a result, in the corollary on p. 121, the 20% becomes 5% (or 1/20 instead of 1/5). (End)
%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).
%H N. J. A. Sloane, <a href="/A001583/b001583.txt">Table of n, a(n) for n = 1..24903</a> (first 1000 terms from T. D. Noe)
%H Bob Bastasz, <a href="https://www.fq.math.ca/Papers1/58-5/bastasz.pdf">Lyndon words of a second-order recurrence</a>, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
%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).
%H Michael J. Mossinghoff and Christopher Pinner, <a href="https://arxiv.org/abs/2205.12439">Prime power order circulant determinants</a>, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
%H H. W. Lloyd Tanner, <a href="https://doi.org/10.1112/plms/s1-18.1.214">On the Binomial Equation x^p-1=0: Quinquisection</a>, Proc. London Math. Soc., 18 (1886-1887), 214-234.
%H H. W. Lloyd Tanner, <a href="https://doi.org/10.1112/plms/s1-24.1.223">On Complex Primes formed with the Fifth Roots of Unity</a>, Proc. London Math. Soc., 24 (1892-1893), 223-262.
%F From _A.H.M. Smeets_, Nov 15 2023: (Start)
%F Equals {prime(m): A296240(m) == 0 (mod 5)}.
%F a(n) ~ prime(20*n). (End)
%t Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* _Jean-François 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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