login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Odd composite integers m such that A004187(m-J(m,45)) == 0 (mod m) and gcd(m,45)=1, where J(m,45) is the Jacobi symbol.
6

%I #11 Dec 29 2020 02:50:59

%S 323,329,377,451,1081,1771,1819,1891,2033,3653,3827,4181,5671,5777,

%T 6601,6721,7471,7931,8149,8557,10877,11309,11663,13201,13861,13981,

%U 14701,15251,15449,17119,17513,17687,17711,17941,18407,19043,19951,20447,20473,23407,23771,23851,23999

%N Odd composite integers m such that A004187(m-J(m,45)) == 0 (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 the identity

%C U(p-J(p,D)) == 0 (mod p) when p is prime, b=1 and D=a^2-4.

%C This sequence contains the odd composite integers with U(m-J(m,D)) == 0 (mod m).

%C For a=7 and b=1, we have D=45 and U(m) recovers 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, 25000, 2], CoprimeQ[#, 45] && CompositeQ[#] && Divisible[ChebyshevU[# - JacobiSymbol[#, 45] - 1, 7/2], #] &]

%Y Cf. A004187, A071904, A081264 (a=1, b=-1), A327653 (a=3,b=-1), A340095 (a=5, b=-1), A340096 (a=7, b=-1), A340097 (a=3, b=1), A340098 (a=5, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Dec 28 2020