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A327652 Intersection of A099011 and A327651.

%I A327652 #25 May 29 2025 06:05:09
%S A327652 169,385,961,1121,3827,6265,6441,6601,7801,8119,10945,13067,15841,
%T A327652 18241,19097,20833,24727,27971,29953,31417,34561,35459,37345,38081,
%U A327652 39059,42127,45961,47321,49105,52633,53041,55969,56953,58241,62481,74305,79361,81361,84587,86033,86241,101311,107801
%N A327652 Intersection of A099011 and A327651.
%C A327652 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 A327652 If k is not required to be coprime to m^2 + 4 (= 8), then there are 1190 such k <= 10^5 and 4847 such k <= 10^6, while there are only 41 terms <= 10^5 and 119 terms <= 10^6 in this sequence.
%H A327652 Daniel Suteu, <a href="/A327652/b327652.txt">Table of n, a(n) for n = 1..10000</a>
%e A327652 169 divides Pell(168) as well as Pell(169) - 1, so 169 is a term.
%o A327652 (PARI) pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
%o A327652 isA327652(n)=!isprime(n) && pellmod(n, n)==kronecker(8,n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1
%Y A327652              m                       m=1           m=2       m=3
%Y A327652 k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  A327651   A327653
%Y A327652 k | x(k)-Kronecker(m^2+4,k)        A049062       A099011   A327654
%Y A327652             both                   A212424       this seq  A327655
%Y A327652 * k is composite and coprime to m^2 + 4.
%Y A327652 Cf. A000129, A091337 ({Kronecker(8,n)}).
%K A327652 nonn
%O A327652 1,1
%A A327652 _Jianing Song_, Sep 20 2019