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 A064987 a(n) = n*sigma(n). 56
 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)=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). 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 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) MAPLE with(numtheory): [n*sigma(n)\$n=1..50]; # Muniru A Asiru, Jan 01 2019 MATHEMATICA # DivisorSigma[1, #]&/@Range  (* 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: A339472 A348034 A086792 * A057341 A068412 A183026 Adjacent sequences:  A064984 A064985 A064986 * A064988 A064989 A064990 KEYWORD mult,nonn,easy,changed AUTHOR Vladeta Jovovic, Oct 30 2001 STATUS approved

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Last modified October 19 09:36 EDT 2021. Contains 348074 sequences. (Running on oeis4.)