cp's OEIS Frontend

A327655 Intersection of A327653 and A327654.

%I A327655 #19 May 29 2025 06:05:21
%S A327655 119,649,1189,4187,12871,14041,16109,23479,24769,28421,31631,34997,
%T A327655 38503,41441,48577,50545,56279,58081,59081,61447,75077,91187,95761,
%U A327655 96139,116821,127937,146329,148943,150281,157693,170039,180517,188501,207761,208349,244649,281017,311579,316409
%N A327655 Intersection of A327653 and A327654.
%C A327655 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 A327655 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 A327655 119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.
%o A327655 (PARI) seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
%o A327655 isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && !seqmod(n-kronecker(13,n), n) && gcd(n,13)==1 && n>1
%Y A327655              m                      m=1            m=2      m=3
%Y A327655 k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  A327653
%Y A327655 k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  A327654
%Y A327655             both                   A212424       A327652  this seq
%Y A327655 * k is composite and coprime to m^2 + 4.
%Y A327655 Cf. A006190, A011583 ({Kronecker(13,n)}).
%K A327655 nonn
%O A327655 1,1
%A A327655 _Jianing Song_, Sep 20 2019