cp's OEIS Frontend

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.

%I A340241 #8 Jan 04 2021 06:29:36
%S A340241 161,323,329,341,377,451,671,901,1007,1079,1081,1271,1819,1853,1891,
%T A340241 2033,2071,2209,2407,2461,2501,2743,3653,3827,4181,4843,5473,5611,
%U A340241 5671,5777,6119,6601,6721,7429,7567,7721,8149,8399,8473,8557,9821,9881,10207,10877,11041,11207,11309,11663
%N 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.
%C A340241 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 A340241 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 A340241 Here b=1, a=7, D=45 and k=3, while U(m) is A004187(m).
%D A340241 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A340241 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
%D A340241 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).
%H A340241 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 A340241 Select[Range[3, 12000, 2], CoprimeQ[#, 45] && CompositeQ[#] &&  Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 45] - 1, 7/2] - 7*JacobiSymbol[#, 45],  #] &]
%Y A340241 Cf. A004187, A071904, A340099 (a=7, b=1, k=1), A340124 (a=7, b=1, k=2).
%Y A340241 Cf. A340239 (a=3, b=1, k=3), A340240 (a=5, b=1, k=3).
%K A340241 nonn
%O A340241 1,1
%A A340241 _Ovidiu Bagdasar_, Jan 01 2021