cp's OEIS Frontend

A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.

%I A327651 #27 May 29 2025 06:04:58
%S A327651 35,169,385,779,899,961,1121,1189,2419,2555,2915,3107,3827,6083,6265,
%T A327651 6441,6601,6895,6965,7801,8119,8339,9179,9809,9881,10403,10763,10835,
%U A327651 10945,13067,14027,14111,15179,15841,18241,18721,19097,20833,20909,22499,23219,24727,26795,27869,27971
%N A327651 Composite numbers k coprime to 8 such that k divides Pell(k - Kronecker(8,k)), Pell = A000129.
%C A327651 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 = 2.
%C A327651 If k is not required to be coprime to m^2 + 4 (= 8), then there are 1232 such k <= 10^5 and 4973 such k <= 10^6, while there are only 83 terms <= 10^5 and 245 terms <= 10^6 in this sequence.
%C A327651 Also composite numbers k coprime to 8 such that A214028(k) divides k - Kronecker(8,k).
%H A327651 Amiram Eldar, <a href="/A327651/b327651.txt">Table of n, a(n) for n = 1..10000</a>
%e A327651 Pell(36) = 21300003689580 is divisible by 35, so 35 is a term.
%o A327651 (PARI) pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
%o A327651 isA327651(n)=!isprime(n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1
%Y A327651              m                       m=1           m=2       m=3
%Y A327651 k | x(k-Kronecker(m^2+4,k))*  A081264 U A141137  this seq  A327653
%Y A327651 k | x(k)-Kronecker(m^2+4,k)        A049062       A099011   A327654
%Y A327651             both                   A212424       A327652   A327655
%Y A327651 * k is composite and coprime to m^2 + 4.
%Y A327651 Cf. A000129, A214028, A091337 ({Kronecker(8,n)}).
%K A327651 nonn
%O A327651 1,1
%A A327651 _Jianing Song_, Sep 20 2019