A327655 Intersection of A327653 and A327654.

119, 649, 1189, 4187, 12871, 14041, 16109, 23479, 24769, 28421, 31631, 34997, 38503, 41441, 48577, 50545, 56279, 58081, 59081, 61447, 75077, 91187, 95761, 96139, 116821, 127937, 146329, 148943, 150281, 157693, 170039, 180517, 188501, 207761, 208349, 244649, 281017, 311579, 316409

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 conditions similar to (a) and (b) hold for k simultaneously, where m = 2.

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.

Examples

			119 divides A006190(120) as well as A006190(119) + 1, so 119 is a term.
		

Crossrefs

m m=1 m=2 m=3

k | x(k-Kronecker(m^2+4,k))* A081264 U A141137 A327651 A327653

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

both A212424 A327652 this seq

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

Cf. A006190, A011583 ({Kronecker(13,n)}).

Programs

PARI

seqmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
isA327655(n)=!isprime(n) && seqmod(n, n)==kronecker(13,n) && !seqmod(n-kronecker(13,n), n) && gcd(n,13)==1 && n>1