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a(n) = gcd(2*n, 4^n)^(2*n + 1) mod (2^(2*n + 1) - 1)^2.
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%I #14 Mar 27 2024 10:41:10

%S 0,8,63,128,1534,2048,16383,32768,524285,524288,4194303,8388608,

%T 100663294,134217728,1073741823,2147483648,42949672956,34359738368,

%U 274877906943,549755813888,6597069766654,8796093022208,70368744177663,140737488355328,2251799813685245

%N a(n) = gcd(2*n, 4^n)^(2*n + 1) mod (2^(2*n + 1) - 1)^2.

%F a(2*n) = 2*4^(2*n)*A001511(2*n) - A001511(n) for n >= 1.

%F a(2*n+1) = 4^(2*n + 1)*(A001511(2*n + 1) + 1) for n >= 1.

%p a := n -> modp(igcd(2*n, 4^n)^(2*n + 1), (2^(2*n + 1) - 1)^2):

%p seq(a(n), n = 0..19);

%o (SageMath)

%o def v2(n): return valuation(2*n, 2)

%o def a(n):

%o if n == 0: return 0

%o return 4^n*(v2(n) + 1) if n % 2 else 2*4^n*v2(n) - v2(n//2)

%o print([a(n) for n in range(0, 25)])

%o (PARI) a(n) = lift(Mod(gcd(2*n, 4^n),(2^(2*n + 1) - 1)^2)^(2*n + 1)); \\ _Michel Marcus_, Mar 27 2024

%o (Python)

%o def A371402(n): return ((~n & n-1).bit_length()+2<<(n<<1) if n&1 else ((m:=(~n & n-1).bit_length())+1<<(n<<1)+1)-m) if n else 0 # _Chai Wah Wu_, Mar 27 2024

%Y Cf. A004171, A001511, A013709, A085058.

%Y Cf. A171977, A089080, A123725.

%K nonn

%O 0,2

%A _Peter Luschny_, Mar 26 2024