A174466 a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

1, 7, 10, 31, 16, 70, 22, 111, 64, 112, 34, 310, 40, 154, 160, 351, 52, 448, 58, 496, 220, 238, 70, 1110, 166, 280, 334, 682, 88, 1120, 94, 1023, 340, 364, 352, 1984, 112, 406, 400, 1776, 124, 1540, 130, 1054, 1024, 490, 142, 3510, 316, 1162, 520, 1240

Offset: 1

Author

Paul D. Hanna, Apr 04 2010

Keywords

• nonn
• mult

Comments

Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.

tau(n) = A000005(n) = the number of divisors of n,

and sigma(n) = A000203(n) = sum of divisors of n.

Dirichlet convolution of A038040 and A000203. - R. J. Mathar, Feb 06 2011

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A007425 (tau_3), A034718, A038040, A174465.

Programs

Haskell

a174466 n = sum $ zipWith3 (((*) .) . (*))
                  divs (map a000203 $ reverse divs) (map a000005 divs)
                  where divs = a027750_row n
-- Reinhard Zumkeller, Jan 21 2014

Magma

[&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // Marius A. Burtea, Oct 18 2019

Mathematica

f[p_, e_] := ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 26 2025 *)

PARI

{a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d,0))}

Formula

Logarithmic derivative of A174465.

Dirichlet g.f.: zeta(s)*(zeta(s-1))^3. - R. J. Mathar, Feb 06 2011

a(n) = Sum_{d|n} tau_3(d)*d = Sum_{d|n} A007425(d)*d. - Enrique Pérez Herrero, Jan 17 2013

G.f.: Sum_{k>=1} k*tau_3(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 06 2018

Sum_{k=1..n} a(k) ~ Pi^2*n^2/24 * (log(n)^2 + ((6*g - 1) + 12*z1/Pi^2) * log(n) + (1 - 6*g + 12*g^2 - 12*sg1)/2 + 6*((6*g - 1)*z1 + z2)/Pi^2), where g is the Euler-Mascheroni constant A001620, sg1 is the first Stieltjes constant A082633, z1 = Zeta'(2) = A073002, z2 = Zeta''(2) = A201994. - Vaclav Kotesovec, Feb 02 2019

Multiplicative with a(p^e) = ((e^2+3*e+2)*p^(e+3) - 2*(e^2+4*e+3)*p^(e+2) + (e^2+5*e+6)*p^(e+1) - 2)/(2*(p-1)^3). - Amiram Eldar, May 26 2025