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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. 2

%I #8 Jan 04 2021 06:29:36

%S 161,323,329,341,377,451,671,901,1007,1079,1081,1271,1819,1853,1891,

%T 2033,2071,2209,2407,2461,2501,2743,3653,3827,4181,4843,5473,5611,

%U 5671,5777,6119,6601,6721,7429,7567,7721,8149,8399,8473,8557,9821,9881,10207,10877,11041,11207,11309,11663

%N 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.

%C 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.

%C 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.

%C Here b=1, a=7, D=45 and k=3, while U(m) is A004187(m).

%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.

%D D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).

%D D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

%H Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, <a href="https://doi.org/10.1016/j.ajmsc.2017.06.002">On Fibonacci and Lucas sequences modulo a prime and primality testing</a>, Arab Journal of Mathematical Sciences, 2018, 24(1), 9--15.

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

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

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

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Jan 01 2021

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Last modified April 30 12:47 EDT 2024. Contains 372134 sequences. (Running on oeis4.)