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a(n) = gcd(Lucas(n)-1, Fibonacci(n)-1).
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%I #18 Sep 08 2022 08:45:23

%S 0,2,1,2,2,1,4,2,3,2,22,1,8,2,29,2,42,1,76,2,55,2,398,1,144,2,521,2,

%T 754,1,1364,2,987,2,7142,1,2584,2,9349,2,13530,1,24476,2,17711,2,

%U 128158,1,46368,2,167761,2,242786,1,439204,2,317811,2,2299702,1

%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 = 2, 4, 5, 8, 9, 10, 14, 15, 16, 20, 22, 26, 27, 28, 32, 34, 38, 39, 40, 44, 46, ... - _Vincenzo Librandi_, Dec 24 2015

%e a(15) = 29 since F(15) - 1 = 3*7*29 and L(15) - 1 = 29*49.

%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 Table[GCD[LucasL[n]-1,Fibonacci[n]-1],{n,60}] (* _Harvey P. Dale_, Sep 25 2017 *)

%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, A115312, A115313, A115314.

%K nonn,easy

%O 1,2

%A _Giovanni Resta_, Jan 20 2006