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A001158 sigma_3(n): sum of cubes of divisors of n.
(Formerly M4605 N1964)
76
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 (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 = 6,7,...,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.

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(prod(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 A059376. - Enrique Pérez Herrero, Jan 19 2013

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

CROSSREFS

Cf. A000005, A000203, A001157.

Cf. A051731, A127093.

Cf. A027748, A124010.

Sequence in context: A065959 A226333 A017669 * A171215 A053819 A085292

Adjacent sequences:  A001155 A001156 A001157 * A001159 A001160 A001161

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane, R. K. Guy

STATUS

approved

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Last modified April 30 07:24 EDT 2016. Contains 272220 sequences.