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Composite terms in A128288(n) = A023163(n)/3 for n>1.
2

%I #13 Apr 08 2019 08:33:44

%S 1853,9701,10877,17261,23323,27403,75077,80189,113573,120581,161027,

%T 162133,163059,196877,213749,291941,361397,400987,427549,482677,

%U 635627,667589,941291,1030373,1033997,1140701,1196061,1256293,1751747,1816363,1842581,2288453,2662277

%N Composite terms in A128288(n) = A023163(n)/3 for n>1.

%C 3 divides A023163(n) for n>1. A023163(n) are the numbers n such that Fibonacci(n) == -2 (mod n).

%C Almost all terms of A128288 are prime that belong to A003631 = {2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97} Primes congruent to {2, 3} mod 5; that are also the primes p that divide Fibonacci(p+1).

%C a(3) = 10877 = 73*149 belongs to A069107 Composite n such that n divides Fibonacci(n+1).

%C a(3) = 10877 and a(4) = 17261 belong to A094395 Odd composite n such that n divides Fibonacci(n) + 1.

%H Giovanni Resta, <a href="/A128289/b128289.txt">Table of n, a(n) for n = 1..1000</a>

%e a(1) = A128288(74) = 1853 = 17*109.

%e a(2) = 9701 = 89*109.

%e a(3) = 10877 = 73*149.

%e a(4) = 17261 = 41*421.

%e a(5) = 23323 = 83*281.

%t Do[ f = Mod[ Fibonacci[3n], 3n ]; If[ !PrimeQ[n] && f == 3n-2, Print[ {n, FactorInteger[n]} ]], {n,1,25000} ]

%Y Cf. A128288, A002708, A023172, A023173, A023162, A023163 = numbers n such that Fib(n) == -2 (mod n). Cf. A003631, A069107, A094413, A094395 = Odd composite n such that n divides Fibonacci(n) + 1.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Feb 24 2007

%E Two more terms from _R. J. Mathar_, Oct 08 2007

%E a(9)-a(33) from _Amiram Eldar_, Apr 07 2019