# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a340239 Showing 1-1 of 1 %I A340239 #8 Jan 04 2021 06:29:46 %S A340239 9,49,63,141,161,207,323,341,377,441,671,901,1007,1127,1281,1449,1853, %T A340239 1891,2071,2303,2407,2501,2743,2961,3827,4181,4623,5473,5611,5777, %U A340239 6119,6593,6601,6721,7161,7567,8149,8473,8961,9729,9881 %N A340239 Odd composite integers m such that A001906(3*m-J(m,5)) == 3*J(m,5) (mod m), where J(m,5) is the Jacobi symbol. %C A340239 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 A340239 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 A340239 Here b=1, a=3, D=5 and k=3, while U(m) is A001906(m). %D A340239 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020. %D A340239 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021). %D A340239 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted). %H A340239 Dorin Andrica, Vlad CriÅŸan, 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. %t A340239 Select[Range[3, 10000, 2], CoprimeQ[#, 5] && CompositeQ[#] && Divisible[ ChebyshevU[3*# - JacobiSymbol[#, 5] - 1, 3/2] - 3*JacobiSymbol[#, 5], #] &] %Y A340239 Cf. A001906, A071904, A340097 (a=3, b=1, k=1), A340122 (a=3, b=1, k=2). %Y A340239 Cf. A340240 (a=5, b=1, k=3), A340241 (a=7, b=1, k=3). %K A340239 nonn %O A340239 1,1 %A A340239 _Ovidiu Bagdasar_, Jan 01 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE