A004009 -id:A004009 - cp's OEIS frontend

A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

1, -24, 252, -1472, 4830, -6048, -16744, 84480, -113643, -115920, 534612, -370944, -577738, 401856, 1217160, 987136, -6905934, 2727432, 10661420, -7109760, -4219488, -12830688, 18643272, 21288960, -25499225, 13865712, -73279080, 24647168

Offset: 1

N. J. A. Sloane

Coefficients of the cusp form of weight 12 for the full modular group.
It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
M. J. Hopkins mentions that the only known primes p for which tau(p) == 1 (mod p) are 11, 23 and 691, that it is an open problem to decide if there are infinitely many such p and that no others are known below 35000. Simon Plouffe has now searched up to tau(314747) and found no other examples. - N. J. A. Sloane, Mar 25 2007
Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
With Dedekind's eta function and the discriminant Delta one has eta(z)^24 = Delta(z)/(2*Pi)^12 = Sum_{m >= 1} tau(m)*q^m, with q = exp(2*Pi*i*z), and z in the complex upper half plane, where i is the imaginary unit. Delta is the eigenfunction of the Hecke operator T_n (n >= 1) with eigenvalue tau(n): T_n Delta = tau(n) Delta. From this the formula for tau(m)*tau(n) given below in the formula section follows. See, e.g., the Koecher-Krieg reference, Lemma and Satz, p. 212. Or the Apostol reference, eq. (3) on p. 114 and the first part of section 6.13 on p. 131. - Wolfdieter Lang, Jan 26 2016
For the functional equation satisfied by the Dirichlet series F(s), Re(s) > 7, of a(n) see the Hardy reference, p. 173, (10.9.4). It is (2*Pi)^(-s) * Gamma(s) * F(s) = (2*Pi)^(s-12) * Gamma(12-s) * F(12-s). This is attributed to J. R. Wilton, 1929, on p. 185. - Wolfdieter Lang, Feb 08 2017
Conjecture: |a(n)| with n > 1 can never be a perfect power. This has been verified for n up to 10^6. - Zhi-Wei Sun, Dec 18 2024
Conjecture: The numbers |a(n)| (n = 1,2,3,...) are distinct. This has been verified for the first 10^6 terms. - Zhi-Wei Sun, Dec 21 2024
Conjecture: |a(n)| > 2*n^4 for all n > 2. This has been verified for n = 3..10^6. - Zhi-Wei Sun, Dec 25 2024
Conjecture: a(m)^2 + a(n)^2 can never be a perfect power. This implies Lehmer's conjecture that a(n) is never zero. We have verified that there is no perfect power among a(m)^2 + a(n)^2 with m,n <= 1000 . - Zhi-Wei Sun, Dec 28 2024
Conjecture: The equation |a(m)a(n)| = x^k with m < n, k > 1 and x >= 0 has no solution. This has been verified for m < n <= 5000. - Zhi-Wei Sun, Dec 29 2024
For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - Zhi-Wei Sun, Jan 25 2025

			G.f. = q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 + ...
35328 = (-24)*(-1472) = a(2)*a(4) = a(2*4) + 2^11*a(2*4/4) = 84480 + 2048*(-24) = 35328. See a comment on T_n Delta = tau(n) Delta above. - _Wolfdieter Lang_, Jan 21 2016

Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, lecture X, pp. 161-185.
Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
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).
H. P. F. Swinnerton-Dyer, Congruence properties of tau(n), pp. 289-311 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.

Simon Plouffe, Table of n, a(n) for n = 1..16090
Jennifer S. Balakrishnan, William Craig, and Ken Ono, Variations of Lehmer's Conjecture for Ramanujan's tau-function, arXiv:2005.10345 [math.NT], 2020.
Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, Even values of Ramanujan's tau-function, arXiv:2102.00111 [math.NT], 2021.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary.
Bruce C. Berndt and Ken Ono, Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024.
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
François Brunault, La fonction Tau de Ramanujan. [Wayback Machine link]
Denis Xavier Charles, Computing The Ramanujan Tau Function.
Benoit Cloitre, On the fractal behavior of primes, 2011.
John Cremona, Home page.
Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, Computing Galois representations and equations for modular curves X_H(l), arXiv:1312.6819 [math.NT], 2013-2014.
Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, Computing the coefficients of a modular form, arXiv:math/0605244 [math.NT], 2006-2010.
John A. Ewell, Ramanujan's Tau Function, Proc. Amer. Math. Soc. 128 (2000), 723-726.
John A. Ewell, Ramanujan's Tau Function.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Luis H. Gallardo, On some formulae for Ramanujan's tau function, Rev. Colomb. Matem. 44 (2010) 103-112.
M. Z. Garaev, V. C. Garcia, and S. V. Konyagin, Waring problem with the Ramanujan tau function, Izvestiya: Mathematics, Vol. 72, No 1 (2008), pp. 35-46; arXiv preprint, arXiv:math/0607169 [math.NT], 2006.
Frank Garvan and Michael J. Schlosser, Combinatorial interpretations of Ramanujan's tau function, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; arXiv preprint, arXiv:1606.08037 [math.CO], 2016.
Hansraj Gupta, The Vanishing of Ramanujan's Function(n), Current Science, 17 (1948), p. 180. [Wayback Machine link]
James Lee Hafner and Jeffrey Stopple, A Heat Kernel Associated to Ramanujan's Tau Function, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
Yang-Hui He and John McKay, Moonshine and the Meaning of Life, in: M. Bhargava et al., Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694, American Mathematical Society, 2017; arXiv preprint, arXiv:1408.2083 [math.NT], 2014.
Michael J. Hopkins, Algebraic topology and modular forms, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317; arXiv preprint, arXiv:math/0212397 [math.AT], 2002.
Masanobu Kaneko and Don Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
Jon Keating and Brady Haran, The Key to the Riemann Hypothesis, Numberphile video (2016).
Jerry B. Keiper, Ramanujan's Tau-Dirichlet Series [Dead link?]
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
N. Laptyeva and V. K. Murty, Fourier coefficients of forms of CM-type, Indian Journal of Pure and Applied Mathematics, Volume 45, Issue 5 (October 2014), pp. 747-758.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 (1970), 495-496.
LMFDB, Newform orbit 1.12.a.a
Florian Luca and Igor E. Shparlinski, Arithmetic properties of the Ramanujan function, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 116. No. 1 (2006), pp. 1-8; arXiv preprint, arXiv:math/0607591 [math.NT], 2006.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010), Article 10.7.4.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
Yuri Matiyasevich, Computational rediscovery of Ramanujan's tau numbers, Integers (2018) 18A, Article #A14.
Keith Matthews, Computing Ramanujan's tau function.
Stephen C. Milne, New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.
Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
Pieter Moree, On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions, arXiv:math/0201265 [math.NT], 2002.
M. Ram Murty and V. Kumar Murty, The Ramanujan tau-function, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23.
M. Ram Murty, V. Kumar Murty, and T. N. Shorey, Odd values of the Ramanujan tau-function, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
Douglas Niebur, A formula for Ramanujan's tau-function, Illinois Journal of Mathematics, vol.19, no.3, pp.448-449, (1975). - _Joerg Arndt_, Sep 06 2015
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Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
David A. Steffen, Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n), 1998.
William A. Stein, Database.
Zhi-Wei Sun, Questions on the reciprocals of the values of the tau function, Question 484763 at MathOverflow, December 25, 2024.
Zhi-Wei Sun, Pythagorean triples and Ramanujan's tau function, Question 484936 at MathOverflow, December 28, 2024.
Zhi-Wei Sun, Additive combinatorics for Ramanujan's tau function, Question 485138 at MathOverflow, January 1, 2025.
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Index entries for "core" sequences.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m.
Index entries for sequences related to Chebyshev polynomials.

Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
For a(n) mod N for various values of N see A046694, A098108, A126812-...
Cf. A006352, A099059, A099060, A262339, A292781.
For primes p such that tau(p) == -1 (mod 23) see A106867.
Cf. A010816, A004009. A013973.
Cf. A126832(n) = a(n) mod 5.

using Nemo
function DedekindEta(len, r)
    R, z = PolynomialRing(ZZ, "z")
    e = eta_qexp(r, len, z)
    [coeff(e, j) for j in 0:len - 1] end
RamanujanTauList(len) = DedekindEta(len, 24)
RamanujanTauList(28) |> println # Peter Luschny, Mar 09 2018

M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);

Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* Michael Somos, May 27 2014 */

M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);

CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
(* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* Dean Hickerson, Jan 03 2003 *)
max = 28; g[k_] := -BernoulliB[k]/(2k) + Sum[ DivisorSigma[k - 1, n - 1]*q^(n - 1), {n, 2, max + 1}]; CoefficientList[ Series[ 8000*g[4]^3 - 147*g[6]^2, {q, 0, max}], q] // Rest (* Jean-François Alcover, Oct 10 2012, from modular forms *)
RamanujanTau[Range[40]] (* The function RamanujanTau is now part of Mathematica's core language so there is no longer any need to load NumberTheory`Ramanujan` before using it *) (* Harvey P. Dale, Oct 12 2012 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* Michael Somos, May 27 2014 *)
a[ n_] := With[{t = Log[q] / (2 Pi I)}, SeriesCoefficient[ Series[ DedekindEta[t]^24, {q, 0, n}], {q, 0, n}]]; (* Michael Somos, May 27 2014 *)

{a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};

{a(n) = if( n<1, 0, polcoeff( x * (sum( i=1, (sqrtint( 8*n - 7) + 1) \ 2,(-1)^i * (2*i - 1) * x^((i^2 - i)/2), O(x^n)))^8, n))};

taup(p,e)={
    if(e==1,
        (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
    ,
        my(t=taup(p,1));
        sum(j=0,e\2,
            (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
        )
    )
};
a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
\\ Charles R Greathouse IV, Apr 22 2013

\\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
a(n) = n^4*sigma(n) - 24*sum(k=1, n-1, (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k));
vector(33, n, a(n)) \\ Joerg Arndt, Sep 06 2015

a(n)=ramanujantau(n) \\ Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A000594(n): return n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) # Chai Wah Wu, Nov 08 2022

def s(n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0}
  s
end
def A000594(n)
  ary = [1]
  a = [0] + (1..n - 1).map{|i| s(i)}
  (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
  ary
end
p A000594(100) # Seiichi Manyama, Mar 26 2017

def A000594(n)
  ary = [0, 1]
  (2..n).each{|i|
    s, t, u = 0, 1, 0
    (1..n).each{|j|
      t += 9 * j
      u += j
      break if i <= u
      s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
    }
    ary << s / (i - 1)
  }
  ary[1..-1]
end
p A000594(100) # Seiichi Manyama, Nov 25 2017

CuspForms( Gamma1(1), 12, prec=100).0; # Michael Somos, May 28 2013

list(delta_qexp(100))[1:] # faster Peter Luschny, May 16 2016

G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 04 2011
abs(a(n)) = O(n^(11/2 + epsilon)), abs(a(p)) <= 2 p^(11/2) if p is prime. These were conjectured by Ramanujan and proved by Deligne.
Zagier says: The proof of these formulas, if written out from scratch, has been estimated at 2000 pages; in his book Manin cites this as a probable record for the ratio: "length of proof:length of statement" in the whole of mathematics.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u + 48*v + 4096*w) - v^3. - Michael Somos, Jul 19 2004
G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - Michael Somos, Dec 09 2013
a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - Michael Somos, Apr 17 2015
a(n) = tau(n) (with tau(0) = 0): tau(m)*tau(n) = Sum_{d| gcd(m,n)} d^11*tau(m*n/d^2), for positive integers m and n. If gcd(m,n) = 1 this gives the multiplicativity of tau. See a comment above with the Koecher-Krieg reference, p. 212, eq. (5). - Wolfdieter Lang, Jan 21 2016
Dirichlet series as product: Sum_{n >= 1} a(n)/n^s = Product_{n >= 1} 1/(1 - a(prime(n))/prime(n)^s  + prime(n)^(11-2*s)). See the Mordell link, eq. (2). - Wolfdieter Lang, May 06 2016. See also Hardy, p. 164, eqs. (10.3.1) and (10.3.8). - Wolfdieter Lang, Jan 27 2017
a(n) is multiplicative with a(prime(n)^k) = sqrt(prime(n)^(11))^k*S(k, a(n) / sqrt(prime(n)^(11))), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2, and A076847(n) = a(prime(n)). See A076847 for alpha multiplicativity and examples. - Wolfdieter Lang, May 17 2016. See also Hardy, p. 164, eq. (10.3.6) rewritten in terms of S. - Wolfdieter Lang, Jan 27 2017
G.f. eta(z)^24 (with q = exp(2*Pi*i*z)) also (E_4(q)^3 - E_6(q)^2) / 1728. See the Hardy reference, p. 166, eq. (10.5.3), with Q = E_4 and R = E_6, given in A004009 and A013973, respectively. - Wolfdieter Lang, Jan 30 2017
a(n) (mod 5) == A126832(n).
a(1) = 1, a(n) = -(24/(n-1))*Sum_{k=1..n-1} A000203(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 26 2017
G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Euler Transform of [-24, -24, -24, -24, ...]. - Simon Plouffe, Jun 21 2018
a(n) = n^4*sigma(n)-24*Sum_{k=1..n-1} (35*k^4-52*k^3*n+18*k^2*n^2)*sigma(k)*sigma(n-k). [See Douglas Niebur link]. - Wesley Ivan Hurt, Jul 22 2025

A001158 sigma_3(n): sum of cubes of divisors of n.

Original entry on oeis.org

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

Offset: 1

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

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

			G.f. = x + 9*x^2 + 28*x^3 + 73*x^4 + 126*x^5 + 252*x^6 + 344*x^7 + ...

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).

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, p. 827.
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
D. B. Lahiri, Some arithmetical identities for Ramanujan's and divisor functions, Bulletin of the Australian Mathematical Society, Volume 1, Issue 3 December 1969, pp. 307-314. See Theorem 2 p. 308.
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.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157.
Cf. A051731, A127093.
Cf. A027748, A124010.
Cf. A004009, A064603 (partial sums).

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

[DivisorSigma(3,n): n in [1..40]]; // Bruno Berselli, Apr 10 2013

seq(numtheory:-sigma[3](n),n=1..100); # Robert Israel, Feb 05 2016

Table[DivisorSigma[3,n],{n,100}] (* corrected by T. D. Noe, Mar 22 2009 *)

makelist(divsum(n,3),n,1,100); /* Emanuele Munarini, Mar 26 2011 */

N=99; q='q+O('q^N);
Vec(sum(n=1,N,n^3*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */

{a(n) = if( n<1, 0, sumdiv(n, d, d^3))}; /* Michael Somos, Jan 07 2017 */

from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 3)
print([a(n) for n in range(1, 43)]) # Michael S. Branicky, Jan 09 2021

[sigma(n, 3) for n in range(1, 40)]  # Zerinvary Lajos, Jun 04 2009

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
G.f.: Sum_{n >= 1} x^n*(1 + 4*x^n + x^(2*n))/(1 - x^n)^4. - Peter Bala, Jan 11 2021
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
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

A013973 Expansion of Eisenstein series E_6(q) (alternate convention E_3(q)).

Original entry on oeis.org

1, -504, -16632, -122976, -532728, -1575504, -4058208, -8471232, -17047800, -29883672, -51991632, -81170208, -129985632, -187132176, -279550656, -384422976, -545530104, -715608432, -986161176, -1247954400, -1665307728, -2066980608, -2678616864, -3243917376, -4159663200

Offset: 0

N. J. A. Sloane

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + ...

W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See R(q).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series

Cf. A004009, A008410, A013974, A145095.
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
Cf. A001160, A286346 (eta(q)^24 / eta(q^2)^12), A286399 (eta(q^2)^12 * eta(q^4)^8 / eta(q)^8).

Basis( ModularForms( Gamma1(1), 6), 25); /* Michael Somos, Apr 01 2015 */

E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(6);
# alternative
A013973 := proc(n)
    if n = 0 then
        1;
    else
        -504*numtheory[sigma][5](n) ;
    end if;
end proc:
seq(A013973(n),n=0..10) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, Boole[n == 0], -504 DivisorSigma[ 5, n]]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 33 (t2 + t3) t2 t3 + t3^3], {q, 0, n}]; (* Michael Somos, Apr 21 2013 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) / 2], {q, 0, 2 n}]; (* Michael Somos, Jun 04 2014 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, (e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2) / QPochhammer[ q^2]^12], {q, 0, n}]; (* Michael Somos, Apr 01 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^3 - 3/2 (t2 + t3) t2 t3 + t3^3], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)
terms = 25; E6[x_] = 1-(12/BernoulliB[6])*Sum[k^5*x^k/(1-x^k), {k, terms}]; CoefficientList[E6[x] + O[x]^terms, x] (* Jean-François Alcover, Feb 28 2018 *)

{a(n) = if( n<1, n==0, -504 * sigma( n, 5))};

{a(n) = my(A, A1, A4); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A)^8; A4 = 32 * x * eta(x^4 + A)^8; polcoeff( (A1 + A4) * (A1^2 - 16 * A1 * A4 - 8 * A4^2) / eta(x^2 + A)^12, n))}; /* Michael Somos, Dec 30 2008 */

ModularForms( Gamma1(1), 6, prec=25).0; # Michael Somos, Jun 04 2013

E6(q) = 1 - 504*Sum_{i>=1} sigma_5(i)q^i where sigma_5(n) is A001160, the sum of fifth powers of the divisors of n. It can also be expressed as E6(q) = 1 - 504*Sum_{i>=1} i^5*q^i/(1-q^i). - Gene Ward Smith, Aug 22 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*v - 8*u^2*w - 66*u*v^2 + 592*u*v*w - 512*u*w^2 + 121*v^3 - 4224*v^2*w + 4096*v*w^2. - Michael Somos, Apr 10 2005
Expansion of Ramanujan's function R(q) = 216*g3 (Weierstrass invariant).
Expansion of (eta(q)^8 + 32 * eta(q^4)^8) * (eta(q)^16 - 512 * eta(q)^8 * eta(q^4)^8 - 8192 * eta(q^4)^16) / eta(q^2)^12 in powers of q. - Michael Somos, Dec 30 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = - (t/i)^6 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
E6(q) = eta(q)^24 / eta(q^2)^12 - 480 * eta(q^2)^12 - 16896 * eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 + 8192 * eta(q^4)^24 / eta(q^2)^12. - Seiichi Manyama, May 08 2017

The g.f. is the 12th root of the g.f. of A282019.

It is easy to see that E_2(x)*E_4(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 is always divisible by 3. Hence, E_2(x)*E_4(x) == 1 (mod 72). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x))^(1/12) = 1 + 18*x - 2088*x^2 + 301296*x^3 - 50784174*x^4 + ... has integer coefficients.

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A282020 Coefficients in q-expansion of (E_2^3 - E_2*E_4)/288, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

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A386815 Coefficients in q-expansion of E_2^4 * E_4^2, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A386816 Coefficients in q-expansion of E_2^2 * E_4^3, where E_2 and E_4 are respectively the Eisenstein series A006352 and A004009.

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A386817 Coefficients in q-expansion of E_2^3 * E_4 * E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A049335 Erroneous version of A004009.

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A282017 Coefficients in q-expansion of (E_4 + E_2^2)/2, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009, respectively.

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A341842 Coefficients of the series whose 12th power equals E_2*E_4, where E_2 and E_4 are the Eisenstein series shown in A006352 and A004009.

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A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).

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A001158 sigma_3(n): sum of cubes of divisors of n.

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