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%I #131 Nov 17 2023 11:49:53
%S 1,4,6,12,10,24,14,32,27,40,22,72,26,56,60,80,34,108,38,120,84,88,46,
%T 192,75,104,108,168,58,240,62,192,132,136,140,324,74,152,156,320,82,
%U 336,86,264,270,184,94,480,147,300,204,312,106,432,220,448,228,232,118
%N a(n) = n*d(n), where d(n) = number of divisors of n (A000005).
%C Dirichlet convolution of sigma(n) (A000203) with phi(n) (A000010). - _Michael Somos_, Jun 08 2000
%C Dirichlet convolution of f(n)=n with itself. See the Apostol reference for Dirichlet convolutions. - _Wolfdieter Lang_, Sep 09 2008
%C Sum of all parts of all partitions of n into equal parts. - _Omar E. Pol_, Jan 18 2013
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
%H T. D. Noe, <a href="/A038040/b038040.txt">Table of n, a(n) for n = 1..1000</a>
%H J. Bourgain, S. V. Konyagin and I. E. Shparlinski, <a href="https://doi.org/10.1093/imrn/rnn090">Product sets of rationals, multiplicative translates of subgroups in residue rings and fixed points of the discrete logarithms</a>, Int. Math. Res. Notices, 2008 (2008), Art. ID rnn 090, 1-29.
%H Jean Bourgain, Sergei Konyagin and Igor Shparlinski. <a href="http://arxiv.org/abs/1103.0567">Distribution on elements of cosets of small subgroups and applications</a>, arXiv:1103.0567 [math.NT], Mar 2 2011.
%H Mikhail R. Gabdullin and Vitalii V. Iudelevich, <a href="https://arxiv.org/abs/2201.09287">Numbers of the form kf(k)</a>, arXiv:2201.09287 [math.NT] (2022).
%H Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%H Paul Pollack, <a href="http://www.math.dartmouth.edu/~ppollack/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, p. 147. [Broken link?]
%H Paul Pollack, <a href="https://web.archive.org/web/20130509125813/http://alpha01.dm.unito.it/personalpages/cerruti/ac/notes.pdf">Analytic and Combinatorial Number Theory</a> Course Notes, p. 147.
%F Dirichlet g.f.: zeta(s-1)^2.
%F G.f.: Sum_{n>=1} n*x^n/(1-x^n)^2. - _Vladeta Jovovic_, Dec 30 2001
%F Sum_{k=1..n} sigma(gcd(n, k)). Multiplicative with a(p^e) = (e+1)*p^e. - _Vladeta Jovovic_, Oct 30 2001
%F Equals A127648 * A127093 * the harmonic series, [1/1, 1/2, 1/3, ...]. - _Gary W. Adamson_, May 10 2007
%F Equals row sums of triangle A127528. - _Gary W. Adamson_, May 21 2007
%F a(n) = n*A000005(n) = A066186(n) - n*(A000041(n) - A000005(n)) = A066186(n) - n*A144300(n). - _Omar E. Pol_, Jan 18 2013
%F a(n) = A000203(n) * A240471(n) + A106315(n). - _Reinhard Zumkeller_, Apr 06 2014
%F L.g.f.: Sum_{k>=1} x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 13 2017
%F a(n) = Sum_{d|n} A018804(d). - _Amiram Eldar_, Jun 23 2020
%F a(n) = Sum_{d|n} phi(d)*sigma(n/d). - _Ridouane Oudra_, Jan 21 2021
%F 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
%F 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
%F 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
%F 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
%F Mobius transform of A060640. - _R. J. Mathar_, Feb 07 2023
%e 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
%p with(numtheory): A038040 := n->tau(n)*n;
%t a[n_] := DivisorSigma[0, n]*n; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Sep 03 2012 *)
%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])
%o (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 */
%o (PARI) a(n) = n*numdiv(n); \\ _Michel Marcus_, Oct 24 2020
%o (MuPAD) n*numlib::tau (n)$ n=1..90 // _Zerinvary Lajos_, May 13 2008
%o (Haskell)
%o a038040 n = a000005 n * n -- _Reinhard Zumkeller_, Jan 21 2014
%o (Python)
%o from sympy import divisor_count as d
%o def a(n): return n*d(n)
%o print([a(n) for n in range(1, 60)]) # _Michael S. Branicky_, Mar 15 2022
%Y Cf. A000005, A000010, A000203, A001620, A018804, A029935, A064987, A062952.
%Y Cf. A127648, A127093, A127528.
%Y Cf. A038044, A143127 (partial sums), A328722 (Dirichlet inverse).
%Y Column 1 of A329323.
%K nonn,easy,mult
%O 1,2
%A _Christian G. Bower_
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