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A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.

%I A339125 #13 Jul 08 2021 23:24:31
%S A339125 9,49,121,169,289,361,529,841,961,1127,1369,1681,1849,2209,2809,3481,
%T A339125 3721,3751,4181,4489,4901,4961,5041,5329,5777,6241,6721,6889,7381,
%U A339125 7921,9409,10201,10609,10877,11449,11881,12769,13201,15251,16129,17161,18081,18769,19321
%N A339125 Odd composite integers m such that A000032(m-J(m,5)) == 2*J(m,5) (mod m), where J(m,5) is the Jacobi symbol.
%C A339125 The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy the identity
%C A339125 V(p-J(p,D)) == 2*J(p,D) (mod p) when p is prime, b=-1 and D=a^2+4.
%C A339125 This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).
%C A339125 For a=1 and b=-1, we have D=5 and V(m) recovers A000032(m) (Lucas numbers).
%D A339125 D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
%D A339125 D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021)
%D A339125 D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted)
%H A339125 Dorin Andrica and Ovidiu Bagdasar, <a href="https://doi.org/10.3390/math9080838">On Generalized Lucas Pseudoprimality of Level k</a>, Mathematics (2021) Vol. 9, 838.
%t A339125 Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 5])] - 2*j, #] &] (* _Amiram Eldar_, Nov 26 2020 *)
%Y A339125 Cf. A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339128 (a=7, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).
%K A339125 nonn
%O A339125 1,1
%A A339125 _Ovidiu Bagdasar_, Nov 24 2020