OFFSET
1,1
COMMENTS
Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n) = m*x(n-1) + x(n-2) for k >= 2. For primes p, we have (a) p divides x(p-((m^2+4)/p); (b) x(p) == ((m^2+4)/p) (mod p), where (D/p) is the Kronecker symbol. This sequence gives composite numbers k such that gcd(k, m^2+4) = 1 and that a condition similar to (b) holds for k, where m = 3.
If k is not required to be coprime to m^2 + 4 (= 13), then there are 352 such k <= 10^5, and 1457 such k <= 10^6, while there are only 54 terms <= 10^5 and 148 terms <= 10^6 in this sequence.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..2000
EXAMPLE
A006190(8) = 3927 == Kronecker(13,8) (mod 8), so 8 is a term.
PROG
(PARI) seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327654(n)=!isprime(n) && seqmod(n, n)==kronecker(13, n) && gcd(n, 13)==1 && n>1
CROSSREFS
m m=1 m=2 m=3
* k is composite and coprime to m^2 + 4.
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 20 2019
STATUS
approved