cp's OEIS Frontend

A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.

%I A319168 #15 Jan 26 2024 15:49:26
%S A319168 4181,6721,13201,15251,34561,51841,64079,64681,67861,68251,90061,
%T A319168 96049,97921,118441,146611,163081,186961,197209,219781,252601,254321,
%U A319168 257761,268801,272611,283361,302101,303101,330929,399001,433621,438751,489601,512461,520801
%N A319168 Frobenius pseudoprimes == 1,4 (mod 5) with respect to Fibonacci polynomial x^2 - x - 1.
%C A319168 Complement of A212423 with respect to A212424.
%C A319168 Intersection of A212424 and A047209.
%C A319168 Composite k == 1,4 (mod 5) such that Fibonacci(k) == 1 (mod k) and that k divides Fibonacci(k-1).
%H A319168 Amiram Eldar, <a href="/A319168/b319168.txt">Table of n, a(n) for n = 1..10000</a> (from Dana Jacobsen's site)
%H A319168 Jon Grantham, <a href="http://dx.doi.org/10.1090/S0025-5718-00-01197-2">Frobenius pseudoprimes</a>, Mathematics of Computation 70 (234): 873-891, 2001. doi: 10.1090/S0025-5718-00-01197-2.
%H A319168 Dana Jacobsen, <a href="http://ntheory.org/pseudoprimes.html">Pseudoprime Statistics, Tables, and Data</a>.
%H A319168 A. Rotkiewicz, <a href="http://www.sbc.org.pl/Content/33711/2003_03.pdf">Lucas and Frobenius Pseudoprimes</a>, Annales Mathematicae Silesiane, 17 (2003): 17-39.
%H A319168 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FrobeniusPseudoprime.html">Frobenius Pseudoprime</a>.
%e A319168 4181 = 37*113 is composite, while Fibonacci(4180) == 0 (mod 4181), Fibonacci(4181) == 1 (mod 4181), so 4181 is a term.
%o A319168 (PARI) for(n=2,500000,if(!isprime(n) && (n%5==1||n%5==4) && fibonacci(n-kronecker(5,n))%n==0 && (fibonacci(n)-kronecker(5,n))%n==0, print1(n, ", ")))
%Y A319168 Cf. A047209, A093372, A094394, A094401, A212423, A212424.
%K A319168 nonn
%O A319168 1,1
%A A319168 _Jianing Song_, Sep 12 2018