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%I M4605 N1964 #133 Jan 27 2024 05:25:23
%S 1,9,28,73,126,252,344,585,757,1134,1332,2044,2198,3096,3528,4681,
%T 4914,6813,6860,9198,9632,11988,12168,16380,15751,19782,20440,25112,
%U 24390,31752,29792,37449,37296,44226,43344,55261,50654,61740,61544,73710,68922,86688
%N sigma_3(n): sum of cubes of divisors of n.
%C If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
%C Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6..24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
%C Also the eigenvalues of the Hecke operator T_n for the entire modular normalized Eisenstein form E_4(z) (see A004009): T_n E_4 = a(n) E_4, n >= 1. For the Hecke operator T_n and eigenforms see, e.g., the Koecher-Krieg reference, p. 207, eq. (5) and p. 211, section 4, or the Apostol reference p. 120, eq. (13) and pp. 129 - 133. - _Wolfdieter Lang_, Jan 28 2016
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 120, 129 - 133.
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
%D Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_4(z).
%H T. D. Noe, <a href="/A001158/b001158.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
%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 D. B. Lahiri, <a href="https://doi.org/10.1017/S0004972700042179">Some arithmetical identities for Ramanujan's and divisor functions</a>, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 2 p. 308.
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function.</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - _David W. Wilson_, Aug 01 2001
%F Dirichlet g.f. zeta(s)*zeta(s-3). - _R. J. Mathar_, Mar 04 2011
%F G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - _Benoit Cloitre_, Apr 21 2003
%F Equals A051731 * [1, 8, 27, 64, 125, ...] = A127093 * [1, 4, 9, 16, 25, ...]. - _Gary W. Adamson_, Nov 02 2007
%F L.g.f.: -log(Product_{j>=1} (1-x^j)^(j^2)) = (1/1)*z^1 + (9/2)*z^2 + (28/3)*z^3 + (73/4)*z^4 + ... + (a(n)/n)*z^n + ... - _Joerg Arndt_, Feb 04 2011
%F a(n) = Sum{d|n} tau_{-2}^d*J_3(n/d), where tau_{-2} is A007427 and J_3 is A059376. - _Enrique Pérez Herrero_, Jan 19 2013
%F a(n) = A004009(n)/240. - _Artur Jasinski_, Sep 06 2016. See, e.g., Hardy, p. 166, (10.5.6), with Q = E_4, and with present offset 0. - _Wolfdieter Lang_, Jan 31 2017
%F 8*a(n) = sum of cubes of even divisors of 2*n. - _Wolfdieter Lang_, Jan 07 2017
%F G.f.: Sum_{n >= 1} x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4. - _Peter Bala_, Jan 11 2021
%F Faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3 + ((n + 1)^3 - 3*n^3)*q^n + (4 - 6*n^2)*q^(2*n) + (3*n^3 - (n - 1)^3)*q^(3*n) - n^3*q^(4*n) )/(1 - q^n)^4 - apply the operator x*d/dx three times to equation 5 in Arndt and then set x = 1. - _Peter Bala_, Jan 21 2021
%F a(n) = Sum_{1 <= i, j, k <= n} tau(gcd(i, j, k, n)) = Sum_{d divides n} tau(d)* J_3(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_3(n) = A059376(n). - _Peter Bala_, Jan 22 2024
%e G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ...
%p seq(numtheory:-sigma[3](n),n=1..100); # _Robert Israel_, Feb 05 2016
%t Table[DivisorSigma[3,n],{n,100}] (* corrected by _T. D. Noe_, Mar 22 2009 *)
%o (PARI) N=99; q='q+O('q^N);
%o Vec(sum(n=1,N,n^3*q^n/(1-q^n))) /* _Joerg Arndt_, Feb 04 2011 */
%o (Sage) [sigma(n, 3) for n in range(1, 40)] # _Zerinvary Lajos_, Jun 04 2009
%o (Maxima) makelist(divsum(n,3),n,1,100); /* _Emanuele Munarini_, Mar 26 2011 */
%o (Magma) [DivisorSigma(3,n): n in [1..40]]; // _Bruno Berselli_, Apr 10 2013
%o (Haskell)
%o a001158 n = product $ zipWith (\p e -> (p^(3*e + 3) - 1) `div` (p^3 - 1))
%o (a027748_row n) (a124010_row n)
%o -- _Reinhard Zumkeller_, Jun 30 2013
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* _Michael Somos_, Jan 07 2017 */
%o (Python)
%o from sympy import divisor_sigma
%o def a(n): return divisor_sigma(n, 3)
%o print([a(n) for n in range(1, 43)]) # _Michael S. Branicky_, Jan 09 2021
%Y Cf. A000005, A000203, A001157.
%Y Cf. A051731, A127093.
%Y Cf. A027748, A124010.
%Y Cf. A004009, A064603 (partial sums).
%K nonn,easy,nice,mult
%O 1,2
%A _N. J. A. Sloane_, _R. K. Guy_
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