

A064987


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


44



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
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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
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 nth level of the staircase described in A244580 (see also A237593).  Omar E. Pol, Oct 11 2018


REFERENCES

B. C. Berndt, Ramanujan's theory of thetafunctions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 163. 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. 166167.


LINKS

Harry J. Smith, Table of n, a(n) for n = 1..1000
M. Planat, Twelvedimensional Pauli group contextuality with eleven rays, arXiv:1201.5455 [quantph], 2012.


FORMULA

Multiplicative with a(p^e) = p^e * (p^(e+1)  1) / (p  1).
G.f.: Sum_{n>0} n^2*x^n/(1x^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(s1)*zeta(s2).  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


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

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.
Sequence in context: A032647 A327165 A086792 * A057341 A068412 A183026
Adjacent sequences: A064984 A064985 A064986 * A064988 A064989 A064990


KEYWORD

mult,nonn,easy


AUTHOR

Vladeta Jovovic, Oct 30 2001


STATUS

approved



