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A327652 Intersection of A099011 and A327651. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A156159 A099011 A330276 * A112076 A305055 A069645

Adjacent sequences:  A327649 A327650 A327651 * A327653 A327654 A327655

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 20 2019

STATUS

approved

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Last modified October 23 11:35 EDT 2021. Contains 348211 sequences. (Running on oeis4.)