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A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).

%I A327654 #19 May 29 2025 06:05:17
%S A327654 4,8,9,119,399,649,1023,1179,1189,1199,1881,2703,3519,4081,4187,5151,
%T A327654 7055,7361,10349,12871,13833,14041,15519,16109,18639,22593,23479,
%U A327654 24769,26937,28421,29007,31631,34111,34997,38503,41441,44671,48577,50545,51711,53823,56279,57407,58081,59081
%N A327654 Composite numbers k coprime to 13 such that k divides A006190(k) - Kronecker(13,k).
%C A327654 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.
%C A327654 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.
%H A327654 Amiram Eldar, <a href="/A327654/b327654.txt">Table of n, a(n) for n = 1..2000</a>
%e A327654 A006190(8) = 3927 == Kronecker(13,8) (mod 8), so 8 is a term.
%o A327654 (PARI) seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
%o A327654 isA327654(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && gcd(n,13)==1 && n>1
%Y A327654              m                      m=1            m=2      m=3
%Y A327654 k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651  A327653
%Y A327654 k | x(k)-Kronecker(m^2+4,k)        A049062       A099011  this seq
%Y A327654             both                   A212424       A327652  A327655
%Y A327654 * k is composite and coprime to m^2 + 4.
%Y A327654 Cf. A006190, A011583 ({Kronecker(13,n)}).
%K A327654 nonn
%O A327654 1,1
%A A327654 _Jianing Song_, Sep 20 2019