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

%I A054785 #53 Jan 19 2024 04:54:45
%S A054785 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 A054785 62,56,80,64,60,96,64,64,96,72,96,104,76,80,112,96,84,128,88,96,156,
%U A054785 96,96,128,114,124,144,112,108,160,144,128,160,120,120,192,124,128,208
%N A054785 a(n) = sigma(2n) - sigma(n), where sigma is the sum of divisors of n, A000203.
%C A054785 Sum of divisors of 2*n that do not divide n. - _Franklin T. Adams-Watters_, Oct 04 2018
%C A054785 a(n) = 2*n iff n = 2^k, k >= 0 (A000079). - _Bernard Schott_, Mar 24 2020
%H A054785 Paul Tek, <a href="/A054785/b054785.txt">Table of n, a(n) for n = 1..10000</a>
%H A054785 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 A054785 a(n) = A000203(2n) - A000203(n).
%F A054785 a(n) = 2*A002131(n).
%F A054785 a(2*n) = A000203(n) + A000593(2*n). - _Reinhard Zumkeller_, Apr 23 2008
%F A054785 L.g.f.: -log(EllipticTheta(4,0,x)) = Sum_{ n>0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jul 05 2016
%F A054785 G.f.: Sum_{k>=1} 2*k*x^k/(1 - x^(2*k)). - _Ilya Gutkovskiy_, Oct 23 2018
%F A054785 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). - _Amiram Eldar_, Jan 19 2024
%e A054785 n=9: sigma(18)=18+9+6+3+2+1=39, sigma(9)=9+3+1=13, a(9)=39-13=26.
%p A054785 a:= proc(n) local e;
%p A054785   e:= 2^padic:-ordp(n,2);
%p A054785   2*e*numtheory:-sigma(n/e)
%p A054785 end proc:
%p A054785 map(a, [$1..100]); # _Robert Israel_, Jul 05 2016
%t A054785 Table[DivisorSigma[1,2n]-DivisorSigma[1,n],{n,70}] (* _Harvey P. Dale_, May 11 2014 *)
%t A054785 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 A054785 (PARI) a(n)=sigma(2*n)-sigma(n) \\ _Charles R Greathouse IV_, Feb 13 2013
%o A054785 (Magma) [DivisorSigma(1, 2*n) - DivisorSigma(1, n): n in [1..70]]; _Vincenzo Librandi_, Oct 05 2018
%Y A054785 Cf. A000079, A000203, A000593, A111003, A002131, A320059.
%K A054785 nonn,easy
%O A054785 1,1
%A A054785 _Labos Elemer_, May 22 2000