A328722 -id:A328722 - cp's OEIS frontend

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

Christian G. Bower

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

			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

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 29 ff.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 162.

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.

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.

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

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

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

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

a(n)=if(n<1,0,direuler(p=2,n,1/(1-p*X)^2)[n])

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 */

a(n) = n*numdiv(n); \\ Michel Marcus, Oct 24 2020

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

[n*sigma(n,0) for n in range(1, 60)] # Stefano Spezia, Jul 20 2025

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

A323910 Dirichlet inverse of the deficiency of n, A033879.

Original entry on oeis.org

1, -1, -2, 0, -4, 4, -6, 0, -1, 6, -10, 2, -12, 8, 10, 0, -16, 1, -18, 2, 14, 12, -22, 4, -3, 14, -2, 2, -28, -16, -30, 0, 22, 18, 26, 4, -36, 20, 26, 4, -40, -24, -42, 2, 4, 24, -46, 8, -5, -1, 34, 2, -52, 0, 42, 4, 38, 30, -58, 2, -60, 32, 6, 0, 50, -40, -66, 2, 46, -40, -70, 12, -72, 38, 2, 2, 62, -48, -78, 8, -4, 42, -82, -2, 66, 44, 58, 4, -88, 2, 74, 2

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323910, Sequence Machine.

Cf. A033879, A323911, A323912, A359549 (parity of terms).

Sequences that appear in the convolution formulas: A002033, A008683, A023900, A055615, A046692, A067824, A074206, A174725, A191161, A327960, A328722, A330575, A345182, A349341, A346246, A349387.

b[n_] := 2 n - DivisorSigma[1, n];
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 17 2020 *)

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -sumdiv(n, d, if(dA033879(n) = (2*n-sigma(n));
v323910 = DirInverse(vector(up_to,n,A033879(n)));
A323910(n) = v323910[n];

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA033879(n/d) * a(d).
From Antti Karttunen, Nov 14 2024: (Start)
Following convolution formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly:
a(n) = Sum_{d|n} A046692(d)*A067824(n/d).
a(n) = Sum_{d|n} A055615(d)*A074206(n/d).
a(n) = Sum_{d|n} A023900(d)*A174725(n/d).
a(n) = Sum_{d|n} A008683(d)*A323912(n/d).
a(n) = Sum_{d|n} A191161(d)*A327960(n/d).
a(n) = Sum_{d|n} A328722(d)*A330575(n/d).
a(n) = Sum_{d|n} A345182(d)*A349341(n/d).
a(n) = Sum_{d|n} A346246(d)*A349387(n/d).
a(n) = Sum_{d|n} A002033(d-1)*A055615(n/d).
(End)

A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

Original entry on oeis.org

1, 0, -1, 0, -3, 2, -5, 0, -2, 2, -9, 4, -11, 2, 5, 0, -15, 2, -17, 4, 7, 2, -21, 8, -6, 2, -4, 4, -27, -2, -29, 0, 11, 2, 17, 8, -35, 2, 13, 8, -39, -6, -41, 4, 8, 2, -45, 16, -10, -2, 17, 4, -51, 0, 29, 8, 19, 2, -57, 4, -59, 2, 12, 0, 35, -14, -65, 4, 23, -10, -69, 24, -71, 2, 4, 4, 47, -18, -77, 16, -8, 2, -81, -4, 47, 2, 29, 8, -87, 4

Offset: 1

Antti Karttunen, Feb 12 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Jon Maiga, Computer-generated formulas for A323912, Sequence Machine.
Wikipedia, Dirichlet convolution.

Cf. A083254, A323913.

Sequences that appear in the convolution formulas: A002033, A023900, A046692, A055615, A067824, A074206, A101035, A130054, A174725, A191161, A253249, A323910 (Möbius transform), A328722, A330575.

up_to = 16384;
DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA083254(n) = (2*eulerphi(n)-n);
v323912 = DirInverse(vector(up_to,n,A083254(n)));
A323912(n) = v323912[n];

A083254(n) = (2*eulerphi(n)-n);
memoA323912 = Map();
A323912(n) = if(1==n,1,my(v); if(mapisdefined(memoA323912,n,&v), v, v = -sumdiv(n,d,if(dA083254(n/d)*A323912(d),0)); mapput(memoA323912,n,v); (v))); \\ Antti Karttunen, Nov 22 2024

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA083254(n/d) * a(d).
From Antti Karttunen, Nov 22 2024: (Start)
Following convolution formulas were conjectured for this sequence by Sequence Machine, with each one giving the first 10000 terms correctly. The first one is certainly true, because A083254 is Möbius transform of A033879:
a(n) = Sum_{d|n} A323910(d).
a(n) = Sum_{d|n} A023900(d)*A074206(n/d) = Sum_{d|n} A002033(d-1)*A023900(n/d).
a(n) = Sum_{d|n} A055615(d)*A067824(n/d)
a(n) = Sum_{d|n} A046692(d)*A253249(n/d)
a(n) = Sum_{d|n} A130054(d)*A174725(n/d)
a(n) = Sum_{d|n} A101035(d)*A330575(n/d)
a(n) = Sum_{d|n} A191161(d)*A328722(n/d)
(End)

A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 16, 0, 48, 0, 32, 36, 80, 0, 48, 0, 112, 120, 80, 0, 72, 0, 80, 168, 176, 0, 192, 100, 208, 108, 112, 0, 0, 0, 192, 264, 272, 280, 360, 0, 304, 312, 320, 0, 0, 0, 176, 180, 368, 0, 480, 196, 200, 408, 208, 0, 432, 440, 448, 456, 464, 0, 960, 0, 496, 252, 448, 520, 0, 0, 272, 552, 0, 0, 864, 0, 592, 300, 304, 616

Offset: 1

Antti Karttunen, Aug 18 2021

nonn

No negative terms in range 1 .. 2^20.

Apparently, A030059 gives the positions of all zeros.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000005, A030059, A030229, A038040, A328722, A347093.

up_to = 16384;
A038040(n) = (n*numdiv(n));
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(dA038040(n)));
A328722(n) = v328722[n];
A347094(n) = (A038040(n)+A328722(n));

a(n) = A038040(n) + A328722(n).
For n > 1, a(n) = -Sum_{d|n, 1A038040(d) * A328722(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A038040(A030229(n)).

A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

Original entry on oeis.org

1, 1, 4, 5, 16, 4, 36, 21, 40, 16, 100, 20, 144, 36, 64, 85, 256, 40, 324, 80, 144, 100, 484, 84, 416, 144, 364, 180, 784, 64, 900, 341, 400, 256, 576, 200, 1296, 324, 576, 336, 1600, 144, 1764, 500, 640, 484, 2116, 340, 1800, 416, 1024, 720, 2704, 364, 1600, 756, 1296, 784

Offset: 1

Werner Schulte, Jan 24 2021

Cf. A000290, A001157, A002618, A007433, A018804, A023900, A060640, A328722.

f[p_, e_] := (p^(2*e) - 1)*(p - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 24 2021 *)

Multiplicative with a(1) = 1 and a(p^e) = (p^(2*e)-1) * (p-1) / (p+1) for prime p and e > 0.
Dirichlet convolution of A002618 and A023900.
Dirichlet convolution of A001157 and A328722.
Dirichlet inverse b(n) for n > 0 is multiplicative with b(1) = 1 and b(p^e) = -(p-1)^2 * e * p^(e-1) for prime p and e > 0.
Dirichlet convolution with A060640 equals A007433.
Dirichlet convolution with A018804 equals A000290.
Sum_{k=1..n} a(k) ~ c * n^3, where c = 12*zeta(3)/Pi^4 = 0.148083... . - Amiram Eldar, Oct 16 2022

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

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A323910 Dirichlet inverse of the deficiency of n, A033879.

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A323912 Dirichlet inverse of A083254(n), where A083254(n) = 2*phi(n) - n.

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A347094 Sum of A038040 (convolution of sigma with Euler phi) and its Dirichlet inverse.

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A340850 Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-2) / (zeta(s-1))^2.

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