A339128 - OEIS

%I #10 Dec 07 2020 18:22:36

%S 9,25,49,51,91,121,125,153,169,289,325,361,441,529,625,637,833,841,

%T 867,961,1183,1225,1369,1633,1681,1849,1921,2209,2599,2601,2651,3481,

%U 3721,4225,4489,4625,5041,5125,5329,5537,6241,6889,7225,7267,7497,7921,8125,8281

%N Odd composite integers m such that A086902(m-J(m,53)) == 2*J(m,53) (mod m), where J(m,53) is the Jacobi symbol.

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

%C This sequence has the odd composite integers with V(m-J(m,D)) == 2*J(m,D) (mod m).

%C For a=7 and b=-1, we have D=53 and V(m) recovers A086902(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)

%t Select[Range[3, 10000, 2], CompositeQ[#] && Divisible[LucasL[# - (j = JacobiSymbol[#, 53]), 7] - 2*j, #] &] (* _Amiram Eldar_, Nov 26 2020 *)

%Y Cf. A086902.

%Y Cf. A339125 (a=1, b=-1), A339126 (a=3, b=-1), A339127 (a=5, b=-1), A339129 (a=3, b=1), A339130 (a=5, b=1), A339131 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Nov 24 2020

%E More terms from _Amiram Eldar_, Nov 26 2020