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A053636
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a(n) = Sum_{odd d|n} phi(d)*2^(n/d).
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5
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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 + ...
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MATHEMATICA
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a[ n_] := If[ n < 1, 0, Sum[ Mod[ d, 2] EulerPhi[ d] 2^(n / d), {d, Divisors[ n]}]] (* Michael Somos, May 09 2013 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (d % 2) * eulerphi(d) * 2^(n / d)))} /* Michael Somos, May 09 2013 */
(Haskell)
a053636 0 = 0
a053636 n = sum $ zipWith (*) (map a000010 ods) (map ((2 ^) . (div n)) ods)
where ods = a182469_row n
(Python)
from sympy import totient, divisors
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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