A064987 a(n) = n*sigma(n).

1, 6, 12, 28, 30, 72, 56, 120, 117, 180, 132, 336, 182, 336, 360, 496, 306, 702, 380, 840, 672, 792, 552, 1440, 775, 1092, 1080, 1568, 870, 2160, 992, 2016, 1584, 1836, 1680, 3276, 1406, 2280, 2184, 3600, 1722, 4032, 1892, 3696, 3510, 3312, 2256, 5952

Offset: 1

Author

Vladeta Jovovic, Oct 30 2001

Keywords

• mult
• nonn
• easy

Comments

Dirichlet convolution of sigma_2(n)=A001157(n) with phi(n)=A000010(n). - Vladeta Jovovic, Oct 27 2002

Equals row sums of triangle A143311 and of triangle A143308. - Gary W. Adamson, Aug 06 2008

a(n) is also the sum of all n's present in A244580, or in other words, a(n) is also the volume (or number of cubes) below the terraces of the n-th level of the staircase described in A244580 (see also A237593). - Omar E. Pol, Oct 11 2018

If n is a superperfect number then sigma(n) is a Mersenne prime and a(n) is a perfect number, a(A019279(k)) = A000396(k), k >= 1, assuming there are no odd perfect numbers. - Omar E. Pol, Apr 15 2020

References

• B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054. see page 43.
• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 166-167.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
• Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
• Michel Planat, Twelve-dimensional Pauli group contextuality with eleven rays, arXiv:1201.5455 [quant-ph], 2012.

Crossrefs

Main diagonal of A319073.

Cf. A000203, A038040, A002618, A000010, A001157, A143308, A143311, A004009, A006352, A000594, A126832, A069097 (Mobius transform), A001001 (inverse Mobius transform), A237593, A244580.

Programs

GAP

a:=List([1..50],n->n*Sigma(n));; Print(a); # Muniru A Asiru, Jan 01 2019

Haskell

a064987 n = a000203 n * n  -- Reinhard Zumkeller, Jan 21 2014

Magma

[n*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jan 01 2019

Maple

with(numtheory): [n*sigma(n)$n=1..50]; # Muniru A Asiru, Jan 01 2019

Mathematica

# DivisorSigma[1,#]&/@Range[80]  (* Harvey P. Dale, Mar 12 2011 *)

MuPAD

numlib::sigma(n)*n$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI

{a(n) = if ( n==0, 0, n * sigma(n))}

PARI

{ for (n=1, 1000, write("b064987.txt", n, " ", n*sigma(n)) ) } \\ Harry J. Smith, Oct 02 2009

Formula

Multiplicative with a(p^e) = p^e * (p^(e+1) - 1) / (p - 1).

G.f.: Sum_{n>0} n^2*x^n/(1-x^n)^2. - Vladeta Jovovic, Oct 27 2002

G.f.: phi_{2, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Michael Somos, Apr 02 2003

G.f. is also (Q - P^2) / 288 where P, Q are Ramanujan Lambert series. - Michael Somos, Apr 02 2003. See the Hardy reference, p. 136, eq. (10.5.4) (with a proof). For Q and P, (10.5.6) and (10.5.5), see E_4 A004009 and E_2 A006352, respectively. - Wolfdieter Lang, Jan 30 2017

Convolution of A000118 and A186690. Dirichlet convolution of A000027 and A000290. - Michael Somos, Mar 25 2012

Dirichlet g.f.: zeta(s-1)*zeta(s-2). - R. J. Mathar, Feb 16 2011

a(n) = A009194(n)*A009242(n). - Michel Marcus, Oct 23 2013

a(n) (mod 5) = A126832(n) = A000594(n) (mod 5). See A126832 for references. - Wolfdieter Lang, Feb 03 2017

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

Sum_{k>=1} 1/a(k) = 1.4383899259334187832765458631783591251241657856627653748389234270650138768... - Vaclav Kotesovec, Sep 20 2020

From Peter Bala, Jan 21 2021: (Start)

G.f.: Sum_{n >= 1} n*q^n*(1 + q^n)/(1 - q^n)^3 (use the expansion x*(1 + x)/(1 - x)^3 = x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + ...).

A faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3*q^(3*n) - (n^3 + 3*n^2 - n)*q^(2*n) - (n^3 - 3*n^2 - n)*q^n + n^3 )/(1 - q^n)^3 - differentiate equation 5 in Arndt w.r.t. both x and q and then set x = 1. (End)

From Richard L. Ollerton, May 07 2021: (Start)

a(n) = Sum_{k=1..n} sigma_2(gcd(n,k)).

a(n) = Sum_{k=1..n} sigma_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

From Peter Bala, Jan 22 2024: (Start)

a(n) = Sum_{1 <= j, k <= n} sigma_1( gcd(j, k, n) ).

a(n) = Sum_{d divides n} sigma_1(d)*J_2(n/d) = Sum_{d divides n} sigma_2(d)* phi(n/d), where the Jordan totient function J_2(n) = A007434(n). (End)