A013973 -id:A013973 - 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.
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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
Oklahoma State Mathematics Department, Ramanujan tau L-Function. [broken link]
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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.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
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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

A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

Original entry on oeis.org

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
E_8 is also the Barnes-Wall lattice in 8 dimensions.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019

			G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
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.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from N. J. A. Sloane)
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
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 Q(q).
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016.
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
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
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.]
Robert V. Moody and Jiri Patera, Fast recursion formula for weight multiplicities, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.
Michael Somos, Introduction to Ramanujan theta functions
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
Martin H. Weissman, Octonions, Cubes, Embeddings, March 2, 2009.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Wikipedia, Eisenstein series
Wikipedia, E_8 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices

Cf. A046948 (partial sums), A000143, A108091 (eighth root).
Cf. A008410, A008411, A001158.
Cf. A006352 (E_2), 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. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8), A035016 (theta_4(q)^8).

Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */

L := Lattice("E",8); A := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */

with(numtheory); 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(4);

a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *)
max = 30; s = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, after Gene Ward Smith *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)

{a(n) = if( n<1, n==0, 240 * sigma(n, 3))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */

q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013

Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006
Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m (A001158).
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
Convolution square is A008410. A008411 is convolution of this sequence with A008410.
Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
G.f. is (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) / 2 where q = exp(Pi i t). So a(n) = A008430(n) + 128*A007331(n) (= A000143(2*n) + 128*A007331(n) = A035016(2*n) + 128*A007331(n)). - Seiichi Manyama, Sep 30 2018
a(n) = 240*A001158(n) if n>0. - Michael Somos, Oct 01 2018
Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - Vaclav Kotesovec, Jan 14 2024

A001160 sigma_5(n), the sum of the 5th powers of the divisors of n.

Original entry on oeis.org

1, 33, 244, 1057, 3126, 8052, 16808, 33825, 59293, 103158, 161052, 257908, 371294, 554664, 762744, 1082401, 1419858, 1956669, 2476100, 3304182, 4101152, 5314716, 6436344, 8253300, 9768751, 12252702, 14408200, 17766056, 20511150

Offset: 1

N. J. A. Sloane

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
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/504. - Simon Plouffe, Mar 01 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
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_6(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 [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.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157, A001158, A001159, A013973, A000584 (Mobius transform), A178448 (Dirichlet inverse)

[DivisorSigma(5,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

A001160 := proc(n)
    numtheory[sigma][5](n);
end proc:
seq(A001160(n),n=1..30) ; # R. J. Mathar, Jan 31 2017

lst={};Do[AppendTo[lst,DivisorSigma[5,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[5,Range[30]] (* Harvey P. Dale, Nov 11 2013 *)

a(n)=sigma(n,5) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n, 5) for n in range(1, 30)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson, Aug 01 2001
G.f.: sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s)*zeta(s-5). - R. J. Mathar, Mar 06 2011
G.f. also (1 - E_6(q))/540, with the g.f. E_6 of A013973. See Hardy p. 166, (10.5.7) with R = E_6. - Wolfdieter Lang, Jan 31 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^4)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
a(n) = Sum_{1 <= i, j, k, l, m <= n} tau(gcd(i, j, k, l, m, n)) = Sum_{d divides n} tau(d) * J_5(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 22 2024

A006352 Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).

Original entry on oeis.org

1, -24, -72, -96, -168, -144, -288, -192, -360, -312, -432, -288, -672, -336, -576, -576, -744, -432, -936, -480, -1008, -768, -864, -576, -1440, -744, -1008, -960, -1344, -720, -1728, -768, -1512, -1152, -1296, -1152, -2184, -912, -1440, -1344, -2160, -1008, -2304, -1056, -2016, -1872, -1728

Offset: 0

N. J. A. Sloane

Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

The series Q(q), R(q) are modular forms, but P(q) is not. - Michael Somos, May 18 2017

			G.f. = 1 - 24*x - 72*x^2 - 96*x^3 - 168*x^4 - 144*x^5 - 288*x^6 + ...

R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see pp. 111 and 113.
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. 19, Eq. (17).

T. D. Noe, Table of n, a(n) for n = 0..1000
F. Beukers, Another congruence for the Apéry numbers, J. Number Theory 25 (1987), no. 2, 201-210.
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms.
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 P(q).
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.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
V. P. Varin, Special solutions to Chazy equation, Comput. Math. Math. Phys. 57, No. 2, 211-235 (2017), eq (75).
Eric Weisstein's World of Mathematics, Eisenstein Series.
Index entries for sequences related to Eisenstein series.

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. A000594 (Delta), A076835, A145155 (Delta').

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(2);

a[n_] := -24*DivisorSigma[1, n]; a[0] = 1; Table[a[n], {n, 0, 46}] (* Jean-François Alcover, Dec 12 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSigma[ 1, n]]; (* Michael Somos, Apr 08 2015 *)

{a(n) = if( n<1, n==0, -24 * sigma(n))}; /* Michael Somos, Apr 09 2003 */

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else -24 * divisor_sigma(n)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

a(n) = -24*sigma(n) = -24*A000203(n), for n>0.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 8*u1*u2 + 6*u1*u3 + 24*u2*u6 - 72*u3*u6. - Michael Somos, May 29 2005
G.f.: 1 - 24*sum(k>=1, k*x^k/(1 - x^k)).
G.f.: 1 + 24 *x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n); (cf. A000203). - Joerg Arndt, Sep 28 2012
G.f.: 1 - 24*x/(1-x) + 48*x^2/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+1)*(k+3)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: q*Delta'/Delta where Delta is the generating function of Ramanujan's tau function (A000594). - Seiichi Manyama, Jul 15 2017

The g.f. is the 12th root of the g.f. of A282102.

It is easy to see that E_2(x)*E_4(x)*E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 + 21*k*5)*x^k/(1 - x^k) (mod 72), and also that the integer k - 10*k^3 + 21*k*5 = k*(3*k^2 - 1)*(7^k^2 - 1) is always divisible by 3. Hence, E_2(x)*E_4(x)*E_6(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)* E_6(x))^(1/12) = 1 - 24*x - 13932*x^2 - 3585216*x^3 - 1580941068*x^4 - ... has integer coefficients.

It is easy to see that E_2(x)^3/E_6(x) == 1 - 72*Sum_{k >= 1} (k - 7*k^5)*x^k/(1 - x^k) (mod 432), and also that the integer k - 7*k^5 is always divisible by 6. Hence, E_2(x)^3/E_6(x) == 1 (mod 432). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)^2/E_6(x))^(1/72) = 1 + 6*x + 1998*x^2 + 722484*x^3 + 291762942* x^4 + ... has integer coefficients.

cp's OEIS Frontend

A341801 Coefficients of the series whose 12th power equals E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.

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A341872 Coefficients of the series whose 72nd power equals E_2(x)^3/E_6(x), where E_2(x) and E_6(x) are the Eisenstein series A006352 and A013973.

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A386814 Coefficients in q-expansion of E_2^4 * E_6, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

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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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A386818 Coefficients in q-expansion of E_2^2 * E_6^2, where E_2 and E_6 are respectively the Eisenstein series A006352 and A013973.

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A198948 Expansion of modular form (E6(q)+8*E6(q^2))/9, where E6 = A013973.

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

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A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

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A341801 Coefficients of the series whose 12th power equals E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973.