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 A001158 sigma_3(n): sum of cubes of divisors of n. (Formerly M4605 N1964) 109
 1, 9, 28, 73, 126, 252, 344, 585, 757, 1134, 1332, 2044, 2198, 3096, 3528, 4681, 4914, 6813, 6860, 9198, 9632, 11988, 12168, 16380, 15751, 19782, 20440, 25112, 24390, 31752, 29792, 37449, 37296, 44226, 43344, 55261, 50654, 61740, 61544, 73710, 68922, 86688 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). 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 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 REFERENCES M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 120, 129 - 133. G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166. Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 207, 211. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). 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). LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827. Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1. Eric Weisstein's World of Mathematics, Divisor Function. FORMULA Multiplicative with a(p^e) = (p^(3e+3)-1)/(p^3-1). - David W. Wilson, Aug 01 2001 Dirichlet g.f. zeta(s)*zeta(s-3). - R. J. Mathar, Mar 04 2011 G.f.: sum(k>=1, k^3*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003 Equals A051731 * [1, 8, 27, 64, 125, ...] = A127093 * [1, 4, 9, 16, 25, ...]. - Gary W. Adamson, Nov 02 2007 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 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 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 8*a(n) = sum of cubes of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017 EXAMPLE G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ... MAPLE seq(numtheory:-sigma[3](n), n=1..100); # Robert Israel, Feb 05 2016 MATHEMATICA Table[DivisorSigma[3, n], {n, 100}] (* corrected by T. D. Noe, Mar 22 2009 *) PROG (PARI) N=99; q='q+O('q^N); Vec(sum(n=1, N, n^3*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */ (Sage) [sigma(n, 3) for n in xrange(1, 40)] # Zerinvary Lajos, Jun 04 2009 (Maxima) makelist(divsum(n, 3), n, 1, 100); /* Emanuele Munarini, Mar 26 2011 */ (MAGMA) [DivisorSigma(3, n): n in [1..40]]; // Bruno Berselli, Apr 10 2013 (Haskell) a001158 n = product \$ zipWith (\p e -> (p^(3*e + 3) - 1) `div` (p^3 - 1))                       (a027748_row n) (a124010_row n) -- Reinhard Zumkeller, Jun 30 2013 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* Michael Somos, Jan 07 2017 */ CROSSREFS Cf. A000005, A000203, A001157. Cf. A051731, A127093. Cf. A027748, A124010. Cf. A004009. Sequence in context: A226333 A017669 A277065 * A171215 A296601 A294567 Adjacent sequences:  A001155 A001156 A001157 * A001159 A001160 A001161 KEYWORD nonn,easy,nice,mult AUTHOR STATUS approved

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Last modified November 16 09:21 EST 2018. Contains 317268 sequences. (Running on oeis4.)