A337782 - OEIS

%I #16 Nov 24 2023 12:06:59

%S 4,8,44,104,136,152,232,286,442,836,1364,1378,2204,2584,2626,2684,

%T 2834,3016,3926,4636,5662,7208,7384,7676,7964,8294,9164,9316,11476,

%U 12524,14824,15224,17324,20026,20474,21736,21944,22814,23804,24616,26596,27028,27404,31124

%N Even composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 7 (mod m), where U(m)=A004187(m) and V(m)=A056854(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=7 and b=1, respectively.

%C For a, b integers, the following sequences are defined:

%C generalized Lucas sequences by U(n+2)=a*U(n+1)-b*U(n) and U(0)=0, U(1)=1,

%C generalized Pell-Lucas sequences by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a.

%C These satisfy the identities U(p)^2 == 1 and V(p)==a (mod p) for p prime and b=1,-1.

%C These numbers may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b. The current sequence is defined for a=7 and b=1.

%H D. Andrica and O. Bagdasar, <a href="https://repository.derby.ac.uk/item/92yqq/on-some-new-arithmetic-properties-of-the-generalized-lucas-sequences">On some new arithmetic properties of the generalized Lucas sequences</a>, preprint for Mediterr. J. Math. 18, 47 (2021).

%t Select[Range[2, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 7/2] - 7, #] && Divisible[ChebyshevU[#-1, 7/2]*ChebyshevU[#-1, 7/2] - 1, #] &]

%Y Cf. A337630 (a=7, b=-1), A337777 (a=3, b=1), A337781 (a=7, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Sep 20 2020

%E More terms from _Amiram Eldar_, Sep 21 2020