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A340099
Odd composite integers m such that A004187(m-J(m,45)) == 0 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.
6
323, 329, 377, 451, 1081, 1771, 1819, 1891, 2033, 3653, 3827, 4181, 5671, 5777, 6601, 6721, 7471, 7931, 8149, 8557, 10877, 11309, 11663, 13201, 13861, 13981, 14701, 15251, 15449, 17119, 17513, 17687, 17711, 17941, 18407, 19043, 19951, 20447, 20473, 23407, 23771, 23851, 23999
OFFSET
1,1
COMMENTS
The generalized Lucas sequences of integer parameters (a,b) defined by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1, satisfy the identity
U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.
This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).
For a=7 and b=1, we have D=45 and U(m) recovers A004187(m).
REFERENCES
D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
LINKS
Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.
MATHEMATICA
Select[Range[3, 25000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 45] - 1, 7/2], #] &]
CROSSREFS
Cf. A004187, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1).
Sequence in context: A033524 A309030 A337781 * A082947 A082948 A182554
KEYWORD
nonn
AUTHOR
Ovidiu Bagdasar, Dec 28 2020
STATUS
approved