login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001583 Artiads: the primes p == 1 (mod 5) for which Fibonacci((p-1)/5) is divisible by p.
(Formerly M5413 N2351)
18
211, 281, 421, 461, 521, 691, 881, 991, 1031, 1151, 1511, 1601, 1871, 1951, 2221, 2591, 3001, 3251, 3571, 3851, 4021, 4391, 4441, 4481, 4621, 4651, 4691, 4751, 4871, 5081, 5281, 5381, 5531, 5591, 5641, 5801, 5881, 6011, 6101, 6211, 6271, 6491, 6841 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
From A.H.M. Smeets, Nov 15 2023: (Start)
Mean gap size between two consecutive terms at p: ~ 20*log(p) (see E. Lehmer).
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)
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..24903 (first 1000 terms from T. D. Noe)
Bob Bastasz, Lyndon words of a second-order recurrence, Fibonacci Quarterly (2020) Vol. 58, No. 5, 25-29.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131.
E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]
E. Lehmer, On the quadratic character of the Fibonacci root, Fib. Quart., 4 (1966), 135-138 (annotated scanned copy).
Michael J. Mossinghoff and Christopher Pinner, Prime power order circulant determinants, arXiv:2205.12439 [math.NT], 2022. See Type 2 primes on p. 3.
H. W. Lloyd Tanner, On the Binomial Equation x^p-1=0: Quinquisection, Proc. London Math. Soc., 18 (1886-1887), 214-234.
H. W. Lloyd Tanner, On Complex Primes formed with the Fifth Roots of Unity, Proc. London Math. Soc., 24 (1892-1893), 223-262.
FORMULA
From A.H.M. Smeets, Nov 15 2023: (Start)
Equals {prime(m): A296240(m) == 0 (mod 5)}.
a(n) ~ prime(20*n). (End)
MATHEMATICA
Select[ Prime[ Range[1000]], Mod[#, 5] == 1 && Divisible[ Fibonacci[(# - 1)/5], #] &] (* Jean-François Alcover, Jun 22 2012 *)
PROG
(Haskell)
a001583 n = a001583_list !! (n-1)
a001583_list = filter
(\p -> mod (a000045 $ div (p - 1) 5) p == 0) a030430_list
-- Reinhard Zumkeller, Aug 15 2013
(PARI) fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
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
CROSSREFS
Cf. A047650, A000045, A024894, subsequence of A030430.
See also A270798 (a subsequence), A270800.
Sequence in context: A346948 A317394 A096706 * A308790 A137870 A137871
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Jan 25 2000
Edited by N. J. A. Sloane, Apr 01 2016
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 18 22:34 EDT 2024. Contains 370951 sequences. (Running on oeis4.)