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A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203. 24
2, 4, 8, 8, 12, 16, 16, 16, 26, 24, 24, 32, 28, 32, 48, 32, 36, 52, 40, 48, 64, 48, 48, 64, 62, 56, 80, 64, 60, 96, 64, 64, 96, 72, 96, 104, 76, 80, 112, 96, 84, 128, 88, 96, 156, 96, 96, 128, 114, 124, 144, 112, 108, 160, 144, 128, 160, 120, 120, 192, 124, 128, 208 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Sum of divisors of 2*n that do not divide n. - Franklin T. Adams-Watters, Oct 04 2018
a(n) = 2*n iff n = 2^k, k >= 0 (A000079). - Bernard Schott, Mar 24 2020
LINKS
Octavio A. Agustín-Aquino, Wang-Sun formula in GL(Z/2kZ), Integers, Vol. 23 (2023), #A37.
FORMULA
a(n) = A000203(2n) - A000203(n).
a(n) = 2*A002131(n).
a(2*n) = A000203(n) + A000593(2*n). - Reinhard Zumkeller, Apr 23 2008
L.g.f.: -log(EllipticTheta(4,0,x)) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
G.f.: Sum_{k>=1} 2*k*x^k/(1 - x^(2*k)). - Ilya Gutkovskiy, Oct 23 2018
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). - Amiram Eldar, Jan 19 2024
EXAMPLE
n=9: sigma(18)=18+9+6+3+2+1=39, sigma(9)=9+3+1=13, a(9)=39-13=26.
MAPLE
a:= proc(n) local e;
e:= 2^padic:-ordp(n, 2);
2*e*numtheory:-sigma(n/e)
end proc:
map(a, [$1..100]); # Robert Israel, Jul 05 2016
MATHEMATICA
Table[DivisorSigma[1, 2n]-DivisorSigma[1, n], {n, 70}] (* Harvey P. Dale, May 11 2014 *)
Table[CoefficientList[Series[-Log[EllipticTheta[4, 0, x]], {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)
PROG
(PARI) a(n)=sigma(2*n)-sigma(n) \\ Charles R Greathouse IV, Feb 13 2013
(Magma) [DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; Vincenzo Librandi, Oct 05 2018
CROSSREFS
Sequence in context: A198346 A078750 A360156 * A236924 A266575 A260514
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, May 22 2000
STATUS
approved

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Last modified July 17 08:16 EDT 2024. Contains 374360 sequences. (Running on oeis4.)