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

0, 2, 4, 12, 16, 40, 72, 140, 256, 540, 1040, 2068, 4128, 8216, 16408, 32880, 65536, 131104, 262296, 524324, 1048640, 2097480, 4194344, 8388652, 16777728, 33554600, 67108912, 134218836, 268435552, 536870968, 1073744160, 2147483708

Offset: 0

Author

N. J. A. Sloane, Mar 23 2000

Keywords

• nonn

Examples

			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 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3321

Crossrefs

Cf. A000010, A001227, A053635, A063776, A182469.

Programs

Haskell

a053636 0 = 0
a053636 n = sum $ zipWith (*) (map a000010 ods) (map ((2 ^) . (div n)) ods)
            where ods = a182469_row n
-- Reinhard Zumkeller, Sep 13 2013

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]] (* Michael Somos, May 09 2013 *)

PARI

{a(n) = if( n<1, 0, sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))} /* Michael Somos, May 09 2013 */

Python

from sympy import totient, divisors
def A053636(n): return (sum(totient(d)<>(~n&n-1).bit_length(),generator=True))<<1) # Chai Wah Wu, Feb 21 2023

Formula

a(n) = n * A063776(n).

a(n) = Sum_{k=1..A001227(n)} A000010(A182469(n,k)) * 2^(n/A182469(n, A001227(n)+1-k)). - Reinhard Zumkeller, Sep 13 2013

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