OFFSET
1,2
COMMENTS
Dirichlet convolution of sigma_2(n)=A001157(n) with phi(n)=A000010(n). - Vladeta Jovovic, Oct 27 2002
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.
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) (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)
MAPLE
with(numtheory): [n*sigma(n)$n=1..50]; # Muniru A Asiru, Jan 01 2019
MATHEMATICA
# DivisorSigma[1, #]&/@Range[80] (* Harvey P. Dale, Mar 12 2011 *)
PROG
(PARI) {a(n) = if ( n==0, 0, n * sigma(n))}
(PARI) { for (n=1, 1000, write("b064987.txt", n, " ", n*sigma(n)) ) } \\ Harry J. Smith, Oct 02 2009
(MuPAD) numlib::sigma(n)*n$ n=1..81 // Zerinvary Lajos, May 13 2008
(Haskell)
a064987 n = a000203 n * n -- Reinhard Zumkeller, Jan 21 2014
(Magma) [n*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jan 01 2019
(GAP) a:=List([1..50], n->n*Sigma(n));; Print(a); # Muniru A Asiru, Jan 01 2019
CROSSREFS
KEYWORD
mult,nonn,easy
AUTHOR
Vladeta Jovovic, Oct 30 2001
STATUS
approved