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

Author

Ovidiu Bagdasar, Jan 01 2021

Keywords

• nonn

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

• 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.

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).

Programs

Mathematica

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