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a(n) = gcd(Lucas(n)-1, Fibonacci(n)+1).
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%I #23 Sep 08 2022 08:45:23

%S 2,2,3,2,2,1,14,2,5,2,18,1,26,2,47,2,34,1,246,2,89,2,322,1,466,2,843,

%T 2,610,1,4414,2,1597,2,5778,1,8362,2,15127,2,10946,1,79206,2,28657,2,

%U 103682,1,150050,2,271443,2,196418,1,1421294,2,514229,2,1860498,1

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

%C Here Lucas is: Lucas(1)=1, Lucas(2)=3 and, for n>2, Lucas(n) = Lucas(n-1) + Lucas(n-2). See A000032.

%C a(n) is prime for n = 1, 2, 3, 4, 5, 8, 9, 10, 14, 15, 16, 20, 21, 22, 26, 28, 32, 33, 34, 38, 40, 44, 45, ... - _Vincenzo Librandi_, Dec 24 2015

%C 350 of the first 1000 terms are primes. - _Harvey P. Dale_, Mar 25 2020

%H Harvey P. Dale, <a href="/A115312/b115312.txt">Table of n, a(n) for n = 1..1000</a>

%e a(15) = 47 since F(15) + 1 =13*47 and L(15) - 1 = 29*47.

%t 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 GCD[#[[1]]-1,#[[2]]+1]&/@With[{nn=60},Thread[{LucasL[Range[ nn]],Fibonacci[ Range[nn]]}]] (* _Harvey P. Dale_, Mar 25 2020 *)

%o (Magma) [Gcd(Lucas(n)-1, Fibonacci(n)+1): n in [1..60]]; // _Vincenzo Librandi_, Dec 24 2015

%o (PARI) a(n) = gcd(fibonacci(n+1)+fibonacci(n-1)-1,fibonacci(n)+1); \\ _Altug Alkan_, Dec 24 2015

%Y Cf. A000032, A000045, A111956, A115311, A115313, A115314.

%K nonn,easy

%O 1,1

%A _Giovanni Resta_, Jan 20 2006