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A371402 a(n) = gcd(2*n, 4^n)^(2*n + 1) mod (2^(2*n + 1) - 1)^2.

%I A371402 #14 Mar 27 2024 10:41:10
%S A371402 0,8,63,128,1534,2048,16383,32768,524285,524288,4194303,8388608,
%T A371402 100663294,134217728,1073741823,2147483648,42949672956,34359738368,
%U A371402 274877906943,549755813888,6597069766654,8796093022208,70368744177663,140737488355328,2251799813685245
%N A371402 a(n) = gcd(2*n, 4^n)^(2*n + 1) mod (2^(2*n + 1) - 1)^2.
%F A371402 a(2*n) = 2*4^(2*n)*A001511(2*n) - A001511(n) for n >= 1.
%F A371402 a(2*n+1) = 4^(2*n + 1)*(A001511(2*n + 1) + 1) for n >= 1.
%p A371402 a := n -> modp(igcd(2*n, 4^n)^(2*n + 1), (2^(2*n + 1) - 1)^2):
%p A371402 seq(a(n), n = 0..19);
%o A371402 (SageMath)
%o A371402 def v2(n): return valuation(2*n, 2)
%o A371402 def a(n):
%o A371402     if n == 0: return 0
%o A371402     return 4^n*(v2(n) + 1) if n % 2 else 2*4^n*v2(n) - v2(n//2)
%o A371402 print([a(n) for n in range(0, 25)])
%o A371402 (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 A371402 (Python)
%o A371402 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 A371402 Cf. A004171, A001511, A013709, A085058.
%Y A371402 Cf. A171977, A089080, A123725.
%K A371402 nonn
%O A371402 0,2
%A A371402 _Peter Luschny_, Mar 26 2024