A038040 a(n) = n*d(n), where d(n) = number of divisors of n (A000005).

1, 4, 6, 12, 10, 24, 14, 32, 27, 40, 22, 72, 26, 56, 60, 80, 34, 108, 38, 120, 84, 88, 46, 192, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 324, 74, 152, 156, 320, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 448, 228, 232, 118

Offset

1,2

Comments

Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - Michael Somos, Jun 08 2000

Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - Wolfdieter Lang, Sep 09 2008

Sum of all parts of all partitions of n into equal parts. - Omar E. Pol, Jan 18 2013

References

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

J. Bourgain, S. V. Konyagin and I. E. Shparlinski, Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.

Jean Bourgain, Sergei Konyagin and Igor Shparlinski. Distribution on elements of cosets of small subgroups and applications, arXiv:1103.0567 [math.NT], Mar 2 2011.

Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).

Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.

Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147. [Broken link?]

Paul Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 147.

Formula

Dirichlet g.f.: zeta(s-1)^2.

G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - Vladeta Jovovic, Dec 30 2001

Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - Vladeta Jovovic, Oct 30 2001

Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, May 10 2007

Equals row sums of triangle A127528. - Gary W. Adamson, May 21 2007

a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - Omar E. Pol, Jan 18 2013

a(n) = A000203(n) * A240471(n) + A106315(n). - Reinhard Zumkeller, Apr 06 2014

L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017

a(n) = Sum_{d|n} A018804(d). - Amiram Eldar, Jun 23 2020

a(n) = Sum_{d|n} phi(d)*sigma(n/d). - Ridouane Oudra, Jan 21 2021

G.f.: Sum_{n >= 1} q^(n^2)*(n^2 + 2*n*q^n - n^2*q^(2*n))/(1 - q^n)^2. - Peter Bala, Jan 22 2021

a(n) = Sum_{k=1..n} sigma(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ x/sqrt(log x). That is, there are 0 < A < B such that Ax/sqrt(log x) < f(x) < Bx/sqrt(log x). - Charles R Greathouse IV, Mar 15 2022

Sum_{k=1..n} a(k) ~ n^2*log(n)/2 + (gamma - 1/4)*n^2, where gamma is Euler's constant (A001620). - Amiram Eldar, Oct 25 2022

Mobius transform of A060640. - R. J. Mathar, Feb 07 2023

Example

For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. The sum of all parts is 6 + 3 + 3 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 = 24 equalling 6 times the number of divisors of 6, so a(6) = 24. - Omar E. Pol, May 08 2021

Maple

with(numtheory): A038040 := n->tau(n)*n;

Mathematica

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 03 2012 *)

Prog

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-p*X)^2)[n])
(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=1, n, k*x^k/(x^k-1)^2, x*O(x^n)), n)) /* Michael Somos, Jan 29 2005 */
(PARI) a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020
(MuPAD) n*numlib::tau (n)$ n=1..90 // Zerinvary Lajos, May 13 2008
(Haskell)
a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014
(Python)
from sympy import divisor_count as d
def a(n): return n*d(n)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Mar 15 2022

Crossrefs

Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.

Cf. A127648, A127093, A127528.

Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).

Column 1 of A329323.

Sequence in context: A189765 A074162 A365648 * A143356 A058270 A332934

Adjacent sequences: A038037 A038038 A038039 * A038041 A038042 A038043

Keyword

nonn,easy,mult

Author

Christian G. Bower

Status

approved