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A053636 a(n) = Sum_{odd d|n} phi(d)*2^(n/d).

%I A053636 #23 Feb 22 2023 01:42:54
%S A053636 0,2,4,12,16,40,72,140,256,540,1040,2068,4128,8216,16408,32880,65536,
%T A053636 131104,262296,524324,1048640,2097480,4194344,8388652,16777728,
%U A053636 33554600,67108912,134218836,268435552,536870968,1073744160,2147483708
%N A053636 a(n) = Sum_{odd d|n} phi(d)*2^(n/d).
%H A053636 Seiichi Manyama, <a href="/A053636/b053636.txt">Table of n, a(n) for n = 0..3321</a>
%F A053636 a(n) = n * A063776(n).
%F A053636 a(n) = Sum_{k=1..A001227(n)} A000010(A182469(n,k)) * 2^(n/A182469(n, A001227(n)+1-k)). - _Reinhard Zumkeller_, Sep 13 2013
%F A053636 G.f.: Sum_{m >= 0} phi(2*m + 1)*2*x^(2*m + 1)/(1 - 2*x^(2*m + 1)). - _Petros Hadjicostas_, Jul 20 2019
%e A053636 2*x + 4*x^2 + 12*x^3 + 16*x^4 + 40*x^5 + 72*x^6 + 140*x^7 + 256*x^8 + 540*x^9 + ...
%t A053636 a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]] (* _Michael Somos_, May 09 2013 *)
%o A053636 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))} /* _Michael Somos_, May 09 2013 */
%o A053636 (Haskell)
%o A053636 a053636 0 = 0
%o A053636 a053636 n = sum $ zipWith (*) (map a000010 ods) (map ((2 ^) . (div n)) ods)
%o A053636             where ods = a182469_row n
%o A053636 -- _Reinhard Zumkeller_, Sep 13 2013
%o A053636 (Python)
%o A053636 from sympy import totient, divisors
%o A053636 def A053636(n): return (sum(totient(d)<<n//d-1 for d in divisors(n>>(~n&n-1).bit_length(),generator=True))<<1) # _Chai Wah Wu_, Feb 21 2023
%Y A053636 Cf. A000010, A001227, A053635, A063776, A182469.
%K A053636 nonn
%O A053636 0,2
%A A053636 _N. J. A. Sloane_, Mar 23 2000