A001583 - OEIS (original) (raw)

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

COMMENTS

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

E. Lehmer, Artiads characterized, J. Math. Anal. Appl. 15 1966 118-131 [annotated and corrected scanned copy]

FORMULA

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

(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