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A146076 Sum of even divisors of n.

%I A146076 #81 Dec 15 2024 05:19:45
%S A146076 0,2,0,6,0,8,0,14,0,12,0,24,0,16,0,30,0,26,0,36,0,24,0,56,0,28,0,48,0,
%T A146076 48,0,62,0,36,0,78,0,40,0,84,0,64,0,72,0,48,0,120,0,62,0,84,0,80,0,
%U A146076 112,0,60,0,144,0,64,0,126,0,96,0,108,0,96,0,182,0,76,0,120,0,112,0,180,0,84,0,192,0,88,0,168,0,156
%N A146076 Sum of even divisors of n.
%C A146076 The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception. A074400 is the main entry.
%C A146076 a(n) is also the total number of parts in all partitions of n into an even number of equal parts. - _Omar E. Pol_, Jun 04 2017
%H A146076 Vincenzo Librandi, <a href="/A146076/b146076.txt">Table of n, a(n) for n = 1..10000</a>
%F A146076 a(2k-1) = 0, a(2k) = 2*sigma(k) for positive k.
%F A146076 Dirichlet g.f.: zeta(s - 1)*zeta(s)*2^(1 - s). - _Geoffrey Critzer_, Mar 29 2015
%F A146076 a(n) = A000203(n) - A000593(n). - _Omar E. Pol_, Apr 05 2016
%F A146076 L.g.f.: -log(Product_{ k>0 } (1-x^(2*k))) = Sum_{ n>=0 } (a(n)/n)*x^n. - _Benedict W. J. Irwin_, Jul 04 2016
%F A146076 a(n) = A000203(n)*(1 - (1/A038712(n))). - _Omar E. Pol_, Aug 01 2018
%F A146076 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - _Amiram Eldar_, Nov 06 2022
%F A146076 Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = 2 - A065442 = 0.393304... . - _Amiram Eldar_, Dec 14 2024
%p A146076 A146076 := proc(n)
%p A146076     if type(n,'even') then
%p A146076         2*numtheory[sigma](n/2) ;
%p A146076     else
%p A146076         0;
%p A146076     end if;
%p A146076 end proc: # _R. J. Mathar_, Dec 07 2017
%t A146076 f[n_] := Plus @@ Select[Divisors[n], EvenQ]; Array[f, 150] (* _Vincenzo Librandi_, May 17 2013 *)
%t A146076 a[n_] := DivisorSum[n, Boole[EvenQ[#]]*#&]; Array[a, 100] (* _Jean-François Alcover_, Dec 01 2015 *)
%t A146076 Table[CoefficientList[Series[-Log[QPochhammer[x^2, x^2]], {x, 0, 60}],x][[n + 1]] n, {n, 1, 60}] (* _Benedict W. J. Irwin_, Jul 04 2016 *)
%t A146076 a[n_] := If[OddQ[n], 0, 2*DivisorSigma[1, n/2]]; Array[a, 100] (* _Amiram Eldar_, Jan 11 2023 *)
%o A146076 (PARI) vector(80, n, if (n%2, 0, sumdiv(n, d, d*(1-(d%2))))) \\ _Michel Marcus_, Mar 30 2015
%o A146076 (PARI) a(n) = if (n%2, 0, 2*sigma(n/2)); \\ _Michel Marcus_, Apr 01 2015
%Y A146076 Cf. A000203, A000593, A006128, A038712, A065442, A074400, A183063, A222171.
%K A146076 nonn,easy
%O A146076 1,2
%A A146076 _N. J. A. Sloane_, Apr 09 2009
%E A146076 Corrected by _Jaroslav Krizek_, May 07 2011