A337781 - OEIS

%I #14 Nov 24 2023 12:06:56

%S 323,329,377,451,1081,1189,1819,1891,2033,2737,2849,3059,3289,3653,

%T 3689,3827,4181,4879,5671,5777,6479,6601,6721,8149,8533,8557,8569,

%U 8651,8701,10199,10877,11309,11339,11521,11663,12341,13201,13489,13861,13981,14701,15251,15301

%N Odd 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[3, 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), A337778 (a=4, b=1), A337779 (a=5, b=1), A337780 (a=6, b=1).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Sep 20 2020