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A064987 a(n) = n*sigma(n). 32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

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

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

M. 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. is 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

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

CROSSREFS

Cf. A000203, A038040, A002618, A000010, A001157, A143308, A143311, A004009, A006352, A000594, A126832.

Sequence in context: A009242 A032647 A086792 * A057341 A068412 A183026

Adjacent sequences:  A064984 A064985 A064986 * A064988 A064989 A064990

KEYWORD

mult,nonn

AUTHOR

Vladeta Jovovic, Oct 30 2001

STATUS

approved

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Last modified February 19 06:46 EST 2018. Contains 299330 sequences. (Running on oeis4.)