OFFSET
1,2
COMMENTS
A Fibonacci based primality criterion of Legendre and Lagrange which is listed as theorem 2.2 in the Zhi-Hong Sun link at A000032 states the Fibonacci(p-1) mod p = (1 - Legendre(p/5))/2. This sequence lists the pseudoprimes to this criterion which are not divisible by 2 or 3.
The number of pseudoprimes appears to decrease as n increases, there being 36 between 1 and 100,000, 17 between 100,000 and 200,000,and 11 between 200,000 and 300,000.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Zhi-Hong Sun, Congruences for Fibonacci Numbers [PDF] (Lecture notes, 2009)
MAPLE
with(numtheory): with(combinat): for n from 1 to 40000 do if n mod 2 <> 0 and n mod 3 <>0 and fibonacci(n-1) mod n = (1-legendre(n, 5))/2 and not isprime(n) then print(n) fi od;
PROG
(PARI) is(n)=gcd(n, 6)==1 && ((Mod([1, 1; 1, 0], n))^(n-1))[1, 2]==(1-kronecker(n, 5))/2 && !isprime(n) \\ Charles R Greathouse IV, Dec 22 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary Detlefs, Dec 09 2012
EXTENSIONS
Missing a(30) added by Charles R Greathouse IV, Dec 25 2012
Name corrected by Jianing Song, Sep 12 2018
STATUS
approved