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

35, 169, 385, 779, 899, 961, 1121, 1189, 2419, 2555, 2915, 3107, 3827, 6083, 6265, 6441, 6601, 6895, 6965, 7801, 8119, 8339, 9179, 9809, 9881, 10403, 10763, 10835, 10945, 13067, 14027, 14111, 15179, 15841, 18241, 18721, 19097, 20833, 20909, 22499, 23219, 24727, 26795, 27869, 27971

Offset: 1

Author

Jianing Song, Sep 20 2019

Keywords

• nonn

Comments

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.

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.

Also composite numbers k coprime to 8 such that A214028(k) divides k - Kronecker(8,k).

Examples

			Pell(36) = 21300003689580 is divisible by 35, so 35 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

m m=1 m=2 m=3

k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 this seq A327653

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

both A212424 A327652 A327655

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

Cf. A000129, A214028, A091337 ({Kronecker(8,n)}).

Programs

PARI

pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327651(n)=!isprime(n) && !pellmod(n-kronecker(8,n), n) && gcd(n,8)==1 && n>1