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A038040 n*d(n), where d(n) = number of divisors of n (A000005). 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

a(n) = n*log(n) + (2G-1)n + O(sqrt(n)), G=eulergamma (Dirichlet).

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

This function appears in an upper bound of fixed points of the discrete logarithms. For a prime p we denote by F(p) the number of solutions of the congruence g^h == h (mod p) for 1 <= g, h <= (p-1). It is noted in [Bourgain et al. (2008), Eq. (33)] that F(p) <= (p-1) tau(p-1) where tau(n) is the number of divisors of n in N. [Jonathan Vos Post, Mar 03 2011].

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

REFERENCES

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

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.

LINKS

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

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

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

P. 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

MAPLE

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

MATHEMATICA

a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* Jean-Fran├žois 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 */

(MuPad)n*numlib::tau (n)$ n=1..90 - Zerinvary Lajos, May 13 2008

(Haskell)

a038040 n = a000005 n * n  -- Reinhard Zumkeller, Jan 21 2014

CROSSREFS

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

Cf. A127648, A127093, A127528.

Cf. A038044, A143127 (partial sums).

Sequence in context: A110758 A189765 A074162 * A143356 A058270 A058199

Adjacent sequences:  A038037 A038038 A038039 * A038041 A038042 A038043

KEYWORD

nonn,easy,mult

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified April 23 15:38 EDT 2017. Contains 285329 sequences.