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A115314 a(n) = gcd(Lucas(n)+1, Fibonacci(n)-1).

%I A115314 #20 Sep 08 2022 08:45:23
%S A115314 2,4,1,2,4,1,6,4,11,2,8,1,58,4,21,2,76,1,110,4,199,2,144,1,1042,4,377,
%T A115314 2,1364,1,1974,4,3571,2,2584,1,18698,4,6765,2,24476,1,35422,4,64079,2,
%U A115314 46368,1,335522,4,121393,2,439204,1,635622,4,1149851,2,832040,1
%N A115314 a(n) = gcd(Lucas(n)+1, Fibonacci(n)-1).
%C A115314 Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1) + Lucas(n-2). See A000032.
%C A115314 a(n) is prime for n = 1, 4, 9, 10, 16, 21, 22, 28, 33, 34, 40, 46, 52, 58, 64, 70, 76, 81, 82, 88, 93, 94, ... - _Vincenzo Librandi_, Dec 24 2015
%H A115314 Harvey P. Dale, <a href="/A115314/b115314.txt">Table of n, a(n) for n = 1..1000</a>
%e A115314 a(15) = 21 = 3*7 since F(15) - 1 = 3*7*29 and L(15) + 1 = 3*5*7*13.
%t A115314 lucas[1]=1; lucas[2]=3; lucas[n_]:= lucas[n]= lucas[n-1] + lucas[n-2]; Table[GCD[lucas[i]+1, Fibonacci[i]-1], {i, 60}]
%t A115314 Module[{nn=60,l,f},l=LucasL[Range[nn]]+1;f=Fibonacci[Range[nn]]-1;GCD@@@ Thread[ {l,f}]] (* _Harvey P. Dale_, Apr 29 2020 *)
%o A115314 (Magma) [Gcd(Lucas(n)+1, Fibonacci(n)-1): n in [1..60]]; // _Vincenzo Librandi_, Dec 24 2015
%o A115314 (PARI) a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)+1,fibonacci(n)-1); \\ _Altug Alkan_, Dec 24 2015
%Y A115314 Cf. A000032, A000045, A111956, A115311, A115312, A115313.
%K A115314 nonn,easy
%O A115314 1,1
%A A115314 _Giovanni Resta_, Jan 20 2006