A340241 Odd composite integers m such that A004187(3*m-J(m,45)) == 7*J(m,45) (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.

161, 323, 329, 341, 377, 451, 671, 901, 1007, 1079, 1081, 1271, 1819, 1853, 1891, 2033, 2071, 2209, 2407, 2461, 2501, 2743, 3653, 3827, 4181, 4843, 5473, 5611, 5671, 5777, 6119, 6601, 6721, 7429, 7567, 7721, 8149, 8399, 8473, 8557, 9821, 9881, 10207, 10877, 11041, 11207, 11309, 11663

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 U(3*p-J(p,D)) == a*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=1 and D=a^2-4.

The composite integers m with the property U(k*m-J(m,D)) == U(k-1)*J(m,D) (mod m) are called generalized Lucas pseudoprimes of level k+ and parameter a.

Here b=1, a=7, D=45 and k=3, while U(m) is 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

Table of n, a(n) for n=1..48.

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, 12000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 45] - 1, 7/2] - 7*JacobiSymbol[#, 45],  #] &]

Crossrefs

Cf. A004187, A071904, A340099 (a=7, b=1, k=1), A340124 (a=7, b=1, k=2).

Cf. A340239 (a=3, b=1, k=3), A340240 (a=5, b=1, k=3).

Sequence in context: A250644 A060641 A209282 * A157954 A159545 A157337

Adjacent sequences: A340238 A340239 A340240 * A340242 A340243 A340244

Keyword

nonn

Author

Ovidiu Bagdasar, Jan 01 2021

Status

approved