A062369 Dirichlet convolution of n and tau^2(n).

1, 6, 7, 21, 9, 42, 11, 58, 30, 54, 15, 147, 17, 66, 63, 141, 21, 180, 23, 189, 77, 90, 27, 406, 54, 102, 106, 231, 33, 378, 35, 318, 105, 126, 99, 630, 41, 138, 119, 522, 45, 462, 47, 315, 270, 162, 51, 987, 86, 324, 147, 357, 57, 636, 135, 638, 161, 198, 63, 1323

Offset

1,2

Comments

Dirichlet convolution of A000027 and A035116.

Inverse Mobius transform of A060724. - R. J. Mathar, Oct 15 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Formula

a(n) = Sum_{i|n, j|n} sigma(i)*sigma(j)/sigma(lcm(i,j)), where sigma(n) = sum of divisors of n.

a(n) = Sum_{i|d, j|d} sigma(gcd(i, j));

a(n) = Sum_{d|n} d*tau(n/d)^2, where tau(n) = number of divisors of n.

Multiplicative with a(p^e) = (1-p^(3+e)-p^(2+e)+e^2+4*p^2+p^2*e^2+2*e-3*p+4*p^2*e-6*e*p-2*e^2*p)/(1-p)^3.

Dirichlet g.f.: (zeta(s))^4*zeta(s-1)/zeta(2*s). - R. J. Mathar, Feb 09 2011

G.f.: Sum_{k>=1} tau(k)^2*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018

Sum_{k=1..n} a(k) ~ 5 * Pi^4 * n^2 / 144. - Vaclav Kotesovec, Jan 28 2019

a(n) = Sum_{d|n} tau(d^2)*sigma(n/d), where tau(n) = number of divisors of n, and sigma(n) = sum of divisors of n. - Ridouane Oudra, Aug 25 2019

Mathematica

a[n_] := Sum[ DivisorSigma[1, i]*DivisorSigma[1, j] / DivisorSigma[1, LCM[i, j]], {i, Divisors[n]}, {j, Divisors[n]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 26 2013 *)

Prog

(PARI) a(n) = sumdiv(n, d, d*numdiv(n/d)^2); \\ Michel Marcus, Nov 03 2018
(Magma) [&+[d*#Divisors(Floor(n/d))^2:d in Divisors(n)]:n in [1..60]]; // Marius A. Burtea, Aug 25 2019

Crossrefs

Cf. A000203, A060648.

Cf. A000005, A060724, A064950, A062369, A062368, A062380.

Sequence in context: A041771 A042757 A257312 * A048062 A295729 A081284

Adjacent sequences: A062366 A062367 A062368 * A062370 A062371 A062372

Keyword

nonn,mult

Author

Vladeta Jovovic, Jul 07 2001

Status

approved