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%I #105 Jan 27 2024 05:28:36
%S 1,6,12,28,30,72,56,120,117,180,132,336,182,336,360,496,306,702,380,
%T 840,672,792,552,1440,775,1092,1080,1568,870,2160,992,2016,1584,1836,
%U 1680,3276,1406,2280,2184,3600,1722,4032,1892,3696,3510,3312,2256,5952
%N a(n) = n*sigma(n).
%C Dirichlet convolution of sigma_2(n)=A001157(n) with phi(n)=A000010(n). - _Vladeta Jovovic_, Oct 27 2002
%C Equals row sums of triangle A143311 and of triangle A143308. - _Gary W. Adamson_, Aug 06 2008
%C 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
%C 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
%D 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.
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 166-167.
%H Harry J. Smith, <a href="/A064987/b064987.txt">Table of n, a(n) for n = 1..1000</a>
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H Passawan Noppakaew and Prapanpong Pongsriiam, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Pongsriiam/pong43.html">Product of Some Polynomials and Arithmetic Functions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
%H Michel Planat, <a href="http://arxiv.org/abs/1201.5455">Twelve-dimensional Pauli group contextuality with eleven rays</a>, arXiv:1201.5455 [quant-ph], 2012.
%F Multiplicative with a(p^e) = p^e * (p^(e+1) - 1) / (p - 1).
%F G.f.: Sum_{n>0} n^2*x^n/(1-x^n)^2. - _Vladeta Jovovic_, Oct 27 2002
%F 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
%F 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
%F Convolution of A000118 and A186690. Dirichlet convolution of A000027 and A000290. - _Michael Somos_, Mar 25 2012
%F Dirichlet g.f.: zeta(s-1)*zeta(s-2). - _R. J. Mathar_, Feb 16 2011
%F a(n) = A009194(n)*A009242(n). - _Michel Marcus_, Oct 23 2013
%F a(n) (mod 5) = A126832(n) = A000594(n) (mod 5). See A126832 for references. - _Wolfdieter Lang_, Feb 03 2017
%F 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
%F Sum_{k>=1} 1/a(k) = 1.4383899259334187832765458631783591251241657856627653748389234270650138768... - _Vaclav Kotesovec_, Sep 20 2020
%F From _Peter Bala_, Jan 21 2021: (Start)
%F 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 + ...).
%F 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)
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F a(n) = Sum_{k=1..n} sigma_2(gcd(n,k)).
%F a(n) = Sum_{k=1..n} sigma_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%F From _Peter Bala_, Jan 22 2024: (Start)
%F a(n) = Sum_{1 <= j, k <= n} sigma_1( gcd(j, k, n) ).
%F 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)
%p with(numtheory): [n*sigma(n)$n=1..50]; # _Muniru A Asiru_, Jan 01 2019
%t # DivisorSigma[1,#]&/@Range[80] (* _Harvey P. Dale_, Mar 12 2011 *)
%o (PARI) {a(n) = if ( n==0, 0, n * sigma(n))}
%o (PARI) { for (n=1, 1000, write("b064987.txt", n, " ", n*sigma(n)) ) } \\ _Harry J. Smith_, Oct 02 2009
%o (MuPAD) numlib::sigma(n)*n$ n=1..81 // _Zerinvary Lajos_, May 13 2008
%o (Haskell)
%o a064987 n = a000203 n * n -- _Reinhard Zumkeller_, Jan 21 2014
%o (Magma) [n*SumOfDivisors(n): n in [1..70]]; // _Vincenzo Librandi_, Jan 01 2019
%o (GAP) a:=List([1..50],n->n*Sigma(n));; Print(a); # _Muniru A Asiru_, Jan 01 2019
%Y Main diagonal of A319073.
%Y Cf. A000203, A038040, A002618, A000010, A001157, A143308, A143311, A004009, A006352, A000594, A126832, A069097 (Mobius transform), A001001 (inverse Mobius transform), A237593, A244580.
%K mult,nonn,easy
%O 1,2
%A _Vladeta Jovovic_, Oct 30 2001
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