%I #14 May 01 2021 05:11:55

%S 10,119,649,1189,1763,3599,4187,5559,6681,12095,12403,12685,12871,

%T 12970,14041,14279,15051,16109,19043,22847,23479,24769,26795,28421,

%U 30743,30889,31631,31647,33919,34997,37949,38503,39203,41441,46079,48577,49141,50523,50545,53301,56279,58081,58589

%N Composite numbers k coprime to 13 such that k divides A006190(k-Kronecker(13,k)).

%C 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 (a) holds for k, where m = 3.

%C If k is not required to be coprime to m^2 + 4 (= 13), then there are 360 such k <= 10^5 and 1506 such k <= 10^6, while there are only 62 terms <= 10^5 and 197 terms <= 10^6 in this sequence.

%C Also composite numbers k coprime to 13 such that A322907(k) divides k - Kronecker(13,k).

%H Amiram Eldar, <a href="/A327653/b327653.txt">Table of n, a(n) for n = 1..10000</a>

%e A006190(9) = 12970 which is divisible by 10, so 10 is a term.

%o (PARI) seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]

%o isA327653(n)=!isprime(n) && !seqmod(n-Kronecker(13,n), n) && gcd(n,13)==1 && n>1

%Y m m=1 m=2 m=3

%Y k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 this seq

%Y k | x(k)-Kronecker(m^2+4,k) A049062 A099011 A327654

%Y both A212424 A327652 A327655

%Y * k is composite and coprime to m^2 + 4.

%Y Cf. A006190, A322907, A011583 ({Kronecker(13,n)}).

%K nonn

%O 1,1

%A _Jianing Song_, Sep 20 2019