A327655 - OEIS

%I #15 Oct 03 2019 09:31:36

%S 119,649,1189,4187,12871,14041,16109,23479,24769,28421,31631,34997,

%T 38503,41441,48577,50545,56279,58081,59081,61447,75077,91187,95761,

%U 96139,116821,127937,146329,148943,150281,157693,170039,180517,188501,207761,208349,244649,281017,311579,316409

%N Intersection of A327653 and A327654.

%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 conditions similar to (a) and (b) hold for k simultaneously, where m = 2.

%C If k is not required to be coprime to m^2 + 4 (= 13), then there are 322 such k <= 10^5 and 1381 such k <= 10^6, while there are only 24 terms <= 10^5 and 72 terms <= 10^6 in this sequence.

%e 119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.

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

%o isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,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  A327653

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

%Y             both                   A212424       A327652  this seq

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

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

%K nonn

%O 1,1

%A _Jianing Song_, Sep 20 2019