This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A174466 #30 May 27 2025 01:08:49
%S A174466 1,7,10,31,16,70,22,111,64,112,34,310,40,154,160,351,52,448,58,496,
%T A174466 220,238,70,1110,166,280,334,682,88,1120,94,1023,340,364,352,1984,112,
%U A174466 406,400,1776,124,1540,130,1054,1024,490,142,3510,316,1162,520,1240
%N A174466 a(n) = Sum_{d|n} d*sigma(n/d)*tau(d).
%C A174466 Compare to sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d) = sum of squares of divisors of n.
%C A174466 tau(n) = A000005(n) = the number of divisors of n,
%C A174466 and sigma(n) = A000203(n) = sum of divisors of n.
%C A174466 Dirichlet convolution of A038040 and A000203. - _R. J. Mathar_, Feb 06 2011
%H A174466 Reinhard Zumkeller, <a href="/A174466/b174466.txt">Table of n, a(n) for n = 1..10000</a>
%F A174466 Logarithmic derivative of A174465.
%F A174466 Dirichlet g.f.: zeta(s)*(zeta(s-1))^3. - _R. J. Mathar_, Feb 06 2011
%F A174466 a(n) = Sum_{d|n} tau_3(d)*d = Sum_{d|n} A007425(d)*d. - _Enrique Pérez Herrero_, Jan 17 2013
%F A174466 G.f.: Sum_{k>=1} k*tau_3(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Sep 06 2018
%F A174466 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
%F A174466 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
%t A174466 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 *)
%o A174466 (PARI) {a(n)=sumdiv(n,d,d*sigma(n/d)*sigma(d,0))}
%o A174466 (Haskell)
%o A174466 a174466 n = sum $ zipWith3 (((*) .) . (*))
%o A174466 divs (map a000203 $ reverse divs) (map a000005 divs)
%o A174466 where divs = a027750_row n
%o A174466 -- _Reinhard Zumkeller_, Jan 21 2014
%o A174466 (Magma) [&+[d*DivisorSigma(1, n div d)*#Divisors(d):d in Divisors(n)]:n in [1..55]]; // _Marius A. Burtea_, Oct 18 2019
%Y A174466 Cf. A000005 (tau), A000203 (sigma), A007425 (tau_3), A034718, A038040, A174465.
%K A174466 nonn,mult
%O A174466 1,2
%A A174466 _Paul D. Hanna_, Apr 04 2010