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Odd composite integers m such that A006497(m) == 3 (mod m).
10

%I #25 Nov 23 2023 11:42:50

%S 33,65,119,273,377,385,533,561,649,1105,1189,1441,2065,2289,2465,2849,

%T 4187,4641,6545,6721,11921,12871,13281,14041,15457,16109,18241,19201,

%U 22049,23479,24769,25345,28421,30745,31631,34997,38121,38503,41441,45961,46761,48577

%N Odd composite integers m such that A006497(m) == 3 (mod m).

%C If p is a prime, then A006497(p) == 3 (mod p).

%C This sequence contains the odd composite integers for which the congruence holds.

%C The generalized Pell-Lucas sequence of integer parameters (a,b) defined by V(n+2)=a*V(n+1)-b*V(n) and V(0)=2, V(1)=a, satisfy the identity V(p)==a (mod p) whenever p is prime and b=-1,1.

%C For a=3, b=-1, V(n) recovers the sequence A006497(n) (bronze Fibonacci numbers).

%D D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

%H Amiram Eldar, <a href="/A335669/b335669.txt">Table of n, a(n) for n = 1..500</a>

%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).

%e 33 is the first odd composite integer for which we have A006497(33) = 132742316047301964 == 3 (mod 33).

%t Select[Range[3, 50000, 2], CompositeQ[#] && Divisible[LucasL[#, 3] - 3, #] &] (* _Amiram Eldar_, Jun 18 2020 *)

%o (PARI) is(m) = m%2 && !isprime(m) && [2, 3]*([0, 1; 1, 3]^m)[, 1]%m==3; \\ _Jinyuan Wang_, Jun 17 2020

%Y Cf. A006497, A005845 (a=1), A330276 (a=2), A335670 (a=4), A335671 (a=5).

%K nonn

%O 1,1

%A _Ovidiu Bagdasar_, Jun 17 2020

%E More terms from _Jinyuan Wang_, Jun 17 2020