A146076 Sum of even divisors of n.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane, Apr 09 2009

Keywords

• nonn
• easy

Comments

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.

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000593, A006128, A038712, A065442, A074400, A183063, A222171.

Programs

Maple

A146076 := proc(n)
    if type(n,'even') then
        2*numtheory[sigma](n/2) ;
    else
        0;
    end if;
end proc: # R. J. Mathar, Dec 07 2017

Mathematica

f[n_] := Plus @@ Select[Divisors[n], EvenQ]; Array[f, 150] (* Vincenzo Librandi, May 17 2013 *)
a[n_] := DivisorSum[n, Boole[EvenQ[#]]*#&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)
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 *)
a[n_] := If[OddQ[n], 0, 2*DivisorSigma[1, n/2]]; Array[a, 100] (* Amiram Eldar, Jan 11 2023 *)

PARI

vector(80, n, if (n%2, 0, sumdiv(n, d, d*(1-(d%2))))) \\ Michel Marcus, Mar 30 2015

PARI

a(n) = if (n%2, 0, 2*sigma(n/2)); \\ Michel Marcus, Apr 01 2015

Formula

a(2k-1) = 0, a(2k) = 2*sigma(k) for positive k.

Dirichlet g.f.: zeta(s - 1)*zeta(s)*2^(1 - s). - Geoffrey Critzer, Mar 29 2015

a(n) = A000203(n) - A000593(n). - Omar E. Pol, Apr 05 2016

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

a(n) = A000203(n)*(1 - (1/A038712(n))). - Omar E. Pol, Aug 01 2018

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 06 2022

Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000203(k) = 2 - A065442 = 0.393304... . - Amiram Eldar, Dec 14 2024

Extensions

Corrected by Jaroslav Krizek, May 07 2011