A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.

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

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

Paul Tek, Table of n, a(n) for n = 1..10000

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

Cf. A000079, A000203, A000593, A111003, A002131, A320059.

Sequence in context: A198346 A078750 A360156 * A236924 A266575 A260514

Adjacent sequences: A054782 A054783 A054784 * A054786 A054787 A054788

Keyword

nonn,easy

Author

Labos Elemer, May 22 2000

Status

approved