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

Author

Labos Elemer, May 22 2000

Keywords

• nonn
• easy

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

Examples

			n=9: sigma(18)=18+9+6+3+2+1=39, sigma(9)=9+3+1=13, a(9)=39-13=26.
		

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.

Crossrefs

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

Programs

Magma

[DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; Vincenzo Librandi, Oct 05 2018

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 *)

PARI

a(n)=sigma(2*n)-sigma(n) \\ Charles R Greathouse IV, Feb 13 2013

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