A327652 Intersection of A099011 and A327651.

169, 385, 961, 1121, 3827, 6265, 6441, 6601, 7801, 8119, 10945, 13067, 15841, 18241, 19097, 20833, 24727, 27971, 29953, 31417, 34561, 35459, 37345, 38081, 39059, 42127, 45961, 47321, 49105, 52633, 53041, 55969, 56953, 58241, 62481, 74305, 79361, 81361, 84587, 86033, 86241, 101311, 107801

Offset

1,1

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 (= 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.

Links

Daniel Suteu, Table of n, a(n) for n = 1..10000

Example

169 divides Pell(168) as well as Pell(169) - 1, so 169 is a term.

Prog

(PARI) pellmod(n, m)=((Mod([2, 1; 1, 0], m))^n)[1, 2]
isA327652(n)=!isprime(n) && pellmod(n, n)==kronecker(8, n) && !pellmod(n-kronecker(8, n), n) && gcd(n, 8)==1 && n>1

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 this seq A327655

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

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

Sequence in context: A099011 A330276 A351337 * A112076 A305055 A069645

Adjacent sequences: A327649 A327650 A327651 * A327653 A327654 A327655

Keyword

nonn

Author

Jianing Song, Sep 20 2019

Status

approved