login
a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.
24

%I #53 Jan 19 2024 04:54:45

%S 2,4,8,8,12,16,16,16,26,24,24,32,28,32,48,32,36,52,40,48,64,48,48,64,

%T 62,56,80,64,60,96,64,64,96,72,96,104,76,80,112,96,84,128,88,96,156,

%U 96,96,128,114,124,144,112,108,160,144,128,160,120,120,192,124,128,208

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

%C Sum of divisors of 2*n that do not divide n. - _Franklin T. Adams-Watters_, Oct 04 2018

%C a(n) = 2*n iff n = 2^k, k >= 0 (A000079). - _Bernard Schott_, Mar 24 2020

%H Paul Tek, <a href="/A054785/b054785.txt">Table of n, a(n) for n = 1..10000</a>

%H Octavio A. Agustín-Aquino, <a href="http://math.colgate.edu/~integers/x37/x37.pdf">Wang-Sun formula in GL(Z/2kZ)</a>, Integers, Vol. 23 (2023), #A37.

%F a(n) = A000203(2n) - A000203(n).

%F a(n) = 2*A002131(n).

%F a(2*n) = A000203(n) + A000593(2*n). - _Reinhard Zumkeller_, Apr 23 2008

%F L.g.f.: -log(EllipticTheta(4,0,x)) = Sum_{ n>0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jul 05 2016

%F G.f.: Sum_{k>=1} 2*k*x^k/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Oct 23 2018

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). - _Amiram Eldar_, Jan 19 2024

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

%p a:= proc(n) local e;

%p e:= 2^padic:-ordp(n,2);

%p 2*e*numtheory:-sigma(n/e)

%p end proc:

%p map(a, [$1..100]); # _Robert Israel_, Jul 05 2016

%t Table[DivisorSigma[1,2n]-DivisorSigma[1,n],{n,70}] (* _Harvey P. Dale_, May 11 2014 *)

%t 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 *)

%o (PARI) a(n)=sigma(2*n)-sigma(n) \\ _Charles R Greathouse IV_, Feb 13 2013

%o (Magma) [DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; _Vincenzo Librandi_, Oct 05 2018

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

%K nonn,easy

%O 1,1

%A _Labos Elemer_, May 22 2000