This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000594 M5153 N2237 #409 Aug 06 2025 12:55:18
%S A000594 1,-24,252,-1472,4830,-6048,-16744,84480,-113643,-115920,534612,
%T A000594 -370944,-577738,401856,1217160,987136,-6905934,2727432,10661420,
%U A000594 -7109760,-4219488,-12830688,18643272,21288960,-25499225,13865712,-73279080,24647168
%N A000594 Ramanujan's tau function (or Ramanujan numbers, or tau numbers).
%C A000594 Coefficients of the cusp form of weight 12 for the full modular group.
%C A000594 It is conjectured that tau(n) is never zero (this has been verified for n < 816212624008487344127999, see the Derickx, van Hoeij, Zeng reference).
%C A000594 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
%C A000594 Number 1 of the 74 eta-quotients listed in Table I of Martin (1996).
%C A000594 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
%C A000594 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
%C A000594 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
%C A000594 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
%C A000594 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
%C A000594 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
%C A000594 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
%C A000594 For some conjectures motivated by additive combinatorics, one may consult the link to Question 485138 at MathOverflow. - _Zhi-Wei Sun_, Jan 25 2025
%D A000594 Tom M. Apostol, Modular functions and Dirichlet series in number theory, second Edition, Springer, 1990, pp. 114, 131.
%D A000594 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000594 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, AMS 2001; see p. 298.
%D A000594 Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (32.2).
%D A000594 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.
%D A000594 Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing of the Ramanujan tau function, preprint, 2001.
%D A000594 Max Koecher and Aloys Krieg, Elliptische Funktionen und Modulformen, 2. Auflage, Springer, 2007, pp. 210 - 212.
%D A000594 Yu. I. Manin, Mathematics and Physics, Birkhäuser, Boston, 1981.
%D A000594 Henry McKean and Victor Moll, Elliptic Curves, Camb. Univ. Press, 1999, p. 139.
%D A000594 M. Ram Murty, The Ramanujan tau-function, pp. 269-288 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D A000594 Srinivasa Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 2000.
%D A000594 Srinivasa Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
%D A000594 Jean-Pierre Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
%D A000594 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, 1994, see p. 482.
%D A000594 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000594 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000594 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.
%D A000594 Don Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt et al., editors, From Number Theory to Physics, Springer-Verlag, 1992.
%D A000594 Don Zagier, "Elliptic modular forms and their applications", in: The 1-2-3 of modular forms, Springer Berlin Heidelberg, 2008, pp. 1-103.
%H A000594 Simon Plouffe, <a href="/A000594/b000594.txt">Table of n, a(n) for n = 1..16090</a>
%H A000594 Jennifer S. Balakrishnan, William Craig, and Ken Ono, <a href="https://arxiv.org/abs/2005.10345">Variations of Lehmer's Conjecture for Ramanujan's tau-function</a>, arXiv:2005.10345 [math.NT], 2020.
%H A000594 Jennifer S. Balakrishnan, Ken Ono, and Wei-Lun Tsai, <a href="https://arxiv.org/abs/2102.00111">Even values of Ramanujan's tau-function</a>, arXiv:2102.00111 [math.NT], 2021.
%H A000594 Bruce C. Berndt and Ken Ono, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s42berndt.pdf">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>.
%H A000594 Bruce C. Berndt and Ken Ono, <a href="http://emis.dsd.sztaki.hu/journals/SLC/wpapers/s42berndt.html">Ramanujan's unpublished manuscript on the partition and tau functions with proofs and commentary</a>, Séminaire Lotharingien de Combinatoire, B42c (1999), 63 pp.
%H A000594 Bruce C. Berndt and Pieter Moree, <a href="https://arxiv.org/abs/2409.03428">Sums of two squares and the tau-function: Ramanujan's trail</a>, arXiv:2409.03428 [math.NT], 2024.
%H A000594 Matthew Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory, Vol. 98, No. 2 (2003), 377-389. MR1955423 (2003k:11071).
%H A000594 François Brunault, <a href="https://web.archive.org/web/20050122020457/http://www.institut.math.jussieu.fr/~brunault/FonctionTau.pdf">La fonction Tau de Ramanujan</a>. [Wayback Machine link]
%H A000594 Denis Xavier Charles, <a href="http://www.cs.wisc.edu/~cdx/CompTau.pdf">Computing The Ramanujan Tau Function</a>.
%H A000594 Benoit Cloitre, <a href="https://web.archive.org/web/20150923071043/http://bcmathematics.monsite-orange.fr/FractalOrderOfPrimes.pdf">On the fractal behavior of primes</a>, 2011.
%H A000594 John Cremona, <a href="https://johncremona.github.io">Home page</a>.
%H A000594 Maarten Derickx, Mark van Hoeij, and Jinxiang Zeng, <a href="https://arxiv.org/abs/1312.6819">Computing Galois representations and equations for modular curves X_H(l)</a>, arXiv:1312.6819 [math.NT], 2013-2014.
%H A000594 Bas Edixhoven, Jean-Marc Couveignes, Robin de Jong, Franz Merkl, and Johan Bosman, <a href="https://arxiv.org/abs/math/0605244">Computing the coefficients of a modular form</a>, arXiv:math/0605244 [math.NT], 2006-2010.
%H A000594 John A. Ewell, <a href="https://dx.doi.org/10.1090/S0002-9939-99-05289-2">Ramanujan's Tau Function</a>, Proc. Amer. Math. Soc. 128 (2000), 723-726.
%H A000594 John A. Ewell, <a href="http://math.la.asu.edu/~rmmc/rmj/Vol28-2/EWE/EWE.html">Ramanujan's Tau Function</a>.
%H A000594 Steven R. Finch, <a href="/A000521/a000521_1.pdf">Modular forms on SL_2(Z)</a>, December 28, 2005. [Cached copy, with permission of the author]
%H A000594 Luis H. Gallardo, <a href="http://www.emis.de/journals/RCM/revistas.art1038.html">On some formulae for Ramanujan's tau function</a>, Rev. Colomb. Matem. 44 (2010) 103-112.
%H A000594 M. Z. Garaev, V. C. Garcia, and S. V. Konyagin, <a href="https://doi.org/10.1070/IM2008v072n01ABEH002390">Waring problem with the Ramanujan tau function</a>, Izvestiya: Mathematics, Vol. 72, No 1 (2008), pp. 35-46; <a href="https://arxiv.org/abs/math/0607169">arXiv preprint</a>, arXiv:math/0607169 [math.NT], 2006.
%H A000594 Frank Garvan and Michael J. Schlosser, <a href="https://doi.org/10.1016/j.disc.2018.07.001">Combinatorial interpretations of Ramanujan's tau function</a>, Discrete Mathematics, Vol. 341, No. 10 (2018), pp. 2831-2840; <a href="https://arxiv.org/abs/1606.08037">arXiv preprint</a>, arXiv:1606.08037 [math.CO], 2016.
%H A000594 Hansraj Gupta, <a href="https://web.archive.org/web/20151210181615/https://www.currentscience.ac.in/Downloads/article_id_017_06_0179_0180_0.pdf">The Vanishing of Ramanujan's Function(n)</a>, Current Science, 17 (1948), p. 180. [Wayback Machine link]
%H A000594 James Lee Hafner and Jeffrey Stopple, <a href="https://doi.org/10.1023/A:1009886102576">A Heat Kernel Associated to Ramanujan's Tau Function</a>, The Ramanujan Journal, Vol. 4, No. 2 (2000), pp. 123-128.
%H A000594 Yang-Hui He and John McKay, <a href="https://doi.org/10.1090/conm/694">Moonshine and the Meaning of Life</a>, in: M. Bhargava et al., Finite Simple Groups: Thirty Years of the Atlas and Beyond, Contemporary Mathematics, Vol. 694, American Mathematical Society, 2017; <a href="https://arxiv.org/abs/1408.2083">arXiv preprint</a>, arXiv:1408.2083 [math.NT], 2014.
%H A000594 Michael J. Hopkins, <a href="https://doi.org/10.1142/4962">Algebraic topology and modular forms</a>, Proc. Internat. Congress Math., Beijing 2002, Vol. I, pp. 291-317; <a href="https://arxiv.org/abs/math/0212397">arXiv preprint</a>, arXiv:math/0212397 [math.AT], 2002.
%H A000594 Masanobu Kaneko and Don Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
%H A000594 Jon Keating and Brady Haran, <a href="https://www.youtube.com/watch?v=VTveQ1ndH1c">The Key to the Riemann Hypothesis</a>, Numberphile video (2016).
%H A000594 Jerry B. Keiper, <a href="http://mathsource.wri.com/MathSource22/Enhancements/NumberTheory/0200-978/Documentation.txt">Ramanujan's Tau-Dirichlet Series</a> [Dead link?]
%H A000594 Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%H A000594 N. Laptyeva and V. K. Murty, <a href="https://doi.org/10.1007/s13226-014-0086-3">Fourier coefficients of forms of CM-type</a>, Indian Journal of Pure and Applied Mathematics, Volume 45, Issue 5 (October 2014), pp. 747-758.
%H A000594 D. H. Lehmer, <a href="http://dx.doi.org/10.1215/S0012-7094-47-01436-1">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433.
%H A000594 D. H. Lehmer, <a href="/A000594/a000594.pdf">The Vanishing of Ramanujan's Function tau(n)</a>, Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
%H A000594 D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-70-99853-4">Tables of Ramanujan's function tau(n)</a>, Math. Comp., 24 (1970), 495-496.
%H A000594 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/1/12/a/a/">Newform orbit 1.12.a.a</a>
%H A000594 Florian Luca and Igor E. Shparlinski, <a href="https://doi.org/10.1007/BF02829735">Arithmetic properties of the Ramanujan function</a>, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 116. No. 1 (2006), pp. 1-8; <a href="https://www.arxiv.org/abs/math/0607591">arXiv preprint</a>, arXiv:math/0607591 [math.NT], 2006.
%H A000594 Nik Lygeros and Olivier Rozier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.html">A new solution to the equation tau(rho) == 0 (mod p)</a>, J. Int. Seq. 13 (2010), Article 10.7.4.
%H A000594 Yves Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc., Vol. 348, No. 12 (1996), 4825-4856, see page 4852 Table I.
%H A000594 Yuri Matiyasevich, <a href="http://math.colgate.edu/~integers/sjs14/sjs14.Abstract.html">Computational rediscovery of Ramanujan's tau numbers</a>, Integers (2018) 18A, Article #A14.
%H A000594 Keith Matthews, <a href="http://www.numbertheory.org/php/tau.html">Computing Ramanujan's tau function</a>.
%H A000594 Stephen C. Milne, <a href="https://doi.org/10.1073/pnas.93.26.15004">New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan's tau function</a>, Proc. Nat. Acad. Sci. USA, 93 (1996) 15004-15008.
%H A000594 Stephen C. Milne, <a href="http://dx.doi.org/10.1023/A:1014865816981">Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions</a>, Ramanujan J., 6 (2002), 7-149.
%H A000594 Louis J. Mordell, <a href="http://www.archive.org/stream/proceedingsofcam1920191721camb#page/n133">On Mr. Ramanujan's empirical expansions of modular functions</a>, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
%H A000594 Pieter Moree, <a href="http://arXiv.org/abs/math.NT/0201265">On some claims in Ramanujan's 'unpublished' manuscript on the partition and tau functions</a>, arXiv:math/0201265 [math.NT], 2002.
%H A000594 M. Ram Murty and V. Kumar Murty, <a href="http://dx.doi.org/10.1007/978-81-322-0770-2_2">The Ramanujan tau-function</a>, in: The mathematical legacy of Srinivasa Ramanujan (Springer, 2012), p 11-23.
%H A000594 M. Ram Murty, V. Kumar Murty, and T. N. Shorey, <a href="http://archive.numdam.org/article/BSMF_1987__115__391_0.pdf">Odd values of the Ramanujan tau-function</a>, Bulletin de la S. M. F., tome 115 (1987), p. 391-395.
%H A000594 Douglas Niebur, <a href="http://projecteuclid.org/euclid.ijm/1256050746">A formula for Ramanujan's tau-function</a>, Illinois Journal of Mathematics, vol.19, no.3, pp.448-449, (1975). - _Joerg Arndt_, Sep 06 2015
%H A000594 Oklahoma State Mathematics Department, <a href="http://www.math.okstate.edu/~loriw/degree2/degree2hm/level1/weight12/weight12.html">Ramanujan tau L-Function</a>. [broken link]
%H A000594 Jon Perry, <a href="https://web.archive.org/web/20030623234353/http://www.users.globalnet.co.uk/~perry/maths/ramanujantau/ramanujantau.htm">Ramanujan's Tau Function</a>. [Wayback Machine link]
%H A000594 Simon Plouffe, <a href="https://web.archive.org/web/20240421145736/https://plouffe.fr/OEIS/b000594.txt">The first 225035 terms</a> (432 MB) [Wayback Machine link]
%H A000594 Simon Plouffe, <a href="http://vixra.org/abs/1409.0048">Conjectures of the OEIS, as of June 20, 2018</a>.
%H A000594 Srinivasa Ramanujan, Collected Papers, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper18/page18.htm">Table of tau(n);n=1 to 30</a>.
%H A000594 Jean-Pierre Serre, <a href="http://www.numdam.org/item/SDPP_1967-1968__9_1_A13_0/">An interpretation of some congruences concerning Ramanujan's tau function</a>, Séminaire Delange-Pisot-Poitou. Théorie des nombres, tome 9, no 1 (1967-1968), exp. no 14, pp. 1-17.
%H A000594 Jean-Pierre Serre, <a href="https://www.researchgate.net/publication/2396158_An_Interpretation_of_some_congruences_concerning_Ramanujan's_tau-function">An interpretation of some congruences concerning Ramanujan's Tau function</a>, 1997.
%H A000594 Jean-Pierre Serre, <a href="https://doi.org/10.1017/S0017089500006194">Sur la lacunarité des puissances de eta</a>, Glasgow Math. Journal, 27 (1985), 203-221.
%H A000594 N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H A000594 N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0207175">My Favorite Integer Sequences</a>, arXiv:math/0207175 [math.CO], 2002.
%H A000594 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>, 2016.
%H A000594 David A. Steffen, <a href="https://les-mathematiques.net/vanilla/uploads/dump_data/2007/0107/08/5466">Les Coefficients de Fourier de la forme modulaire: La fonction de Ramanujan tau(n)</a>, 1998.
%H A000594 William A. Stein, <a href="http://wstein.org/">Database</a>.
%H A000594 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/484763">Questions on the reciprocals of the values of the tau function</a>, Question 484763 at MathOverflow, December 25, 2024.
%H A000594 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/484936">Pythagorean triples and Ramanujan's tau function</a>, Question 484936 at MathOverflow, December 28, 2024.
%H A000594 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/485138">Additive combinatorics for Ramanujan's tau function</a>, Question 485138 at MathOverflow, January 1, 2025.
%H A000594 H. P. F. Swinnerton-Dyer, <a href="http://dx.doi.org/10.1007/978-3-540-37802-0_1">On l-adic representations and congruences for coefficients of modular forms</a>, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
%H A000594 F. Van der Blij, The function tau (n) of S. Ramanujan (an expository lecture), Math. Student, Vol. 18, No. 3 (1950), pp. 83-99; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11720">entire issues 3 and 4</a>.
%H A000594 Jan Vonk, <a href="https://doi.org/10.1090/bull/1700">Overconvergent modular forms and their explicit arithmetic</a>, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
%H A000594 G. N. Watson, <a href="http://dx.doi.org/10.1112/plms/s2-51.1.1">A table of Ramanujan's function tau(n)</a>, Proc. London Math. Soc., 51 (1950), 1-13.
%H A000594 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>.
%H A000594 Kenneth S. Williams, <a href="http://dx.doi.org/10.4169/amer.math.monthly.122.01.30">Historical remark on Ramanujan's tau function</a>, Amer. Math. Monthly, 122 (2015), 30-35; <a href="http://people.math.carleton.ca/~williams/papers/pdf/355.pdf">author's copy</a>.
%H A000594 <a href="/index/Cor#core">Index entries for "core" sequences</a>.
%H A000594 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>.
%H A000594 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%F A000594 G.f.: x * Product_{k>=1} (1 - x^k)^24 = x*A(x)^8, with the g.f. of A010816.
%F A000594 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
%F A000594 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.
%F A000594 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.
%F A000594 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
%F A000594 G.f. A(q) satisfies q * d log(A(q))/dq = A006352(q). - _Michael Somos_, Dec 09 2013
%F A000594 a(2*n) = A099060(n). a(2*n + 1) = A099059(n). - _Michael Somos_, Apr 17 2015
%F A000594 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
%F A000594 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
%F A000594 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
%F A000594 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
%F A000594 a(n) (mod 5) == A126832(n).
%F A000594 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
%F A000594 G.f.: x*exp(-24*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%F A000594 Euler Transform of [-24, -24, -24, -24, ...]. - _Simon Plouffe_, Jun 21 2018
%F A000594 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
%e A000594 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 + ...
%e A000594 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
%p A000594 M := 50; t1 := series(x*mul((1-x^k)^24,k=1..M),x,M); A000594 := n-> coeff(t1,x,n);
%t A000594 CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 30} ]], 30], x] (* Or *)
%t A000594 (* first do *) Needs["NumberTheory`Ramanujan`"] (* then *) Table[ RamanujanTau[n], {n, 30}] (* _Dean Hickerson_, Jan 03 2003 *)
%t A000594 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 *)
%t A000594 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 *)
%t A000594 a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^24, {q, 0, n}]; (* _Michael Somos_, May 27 2014 *)
%t A000594 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 *)
%o A000594 (Julia)
%o A000594 using Nemo
%o A000594 function DedekindEta(len, r)
%o A000594 R, z = PolynomialRing(ZZ, "z")
%o A000594 e = eta_qexp(r, len, z)
%o A000594 [coeff(e, j) for j in 0:len - 1] end
%o A000594 RamanujanTauList(len) = DedekindEta(len, 24)
%o A000594 RamanujanTauList(28) |> println # _Peter Luschny_, Mar 09 2018
%o A000594 (Magma) M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],100); Coefficients($1);
%o A000594 (Magma) Basis( CuspForms( Gamma1(1), 12), 100)[1]; /* _Michael Somos_, May 27 2014 */
%o A000594 (PARI) {a(n) = if( n<1, 0, polcoeff( x * eta(x + x * O(x^n))^24, n))};
%o A000594 (PARI) {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))};
%o A000594 (PARI) taup(p,e)={
%o A000594 if(e==1,
%o A000594 (65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
%o A000594 ,
%o A000594 my(t=taup(p,1));
%o A000594 sum(j=0,e\2,
%o A000594 (-1)^j*binomial(e-j,e-2*j)*p^(11*j)*t^(e-2*j)
%o A000594 )
%o A000594 )
%o A000594 };
%o A000594 a(n)=my(f=factor(n));prod(i=1,#f[,1],taup(f[i,1],f[i,2]));
%o A000594 \\ _Charles R Greathouse IV_, Apr 22 2013
%o A000594 (PARI) \\ compute terms individually (Douglas Niebur, Ill. J. Math., 19, 1975):
%o A000594 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));
%o A000594 vector(33, n, a(n)) \\ _Joerg Arndt_, Sep 06 2015
%o A000594 (PARI) a(n)=ramanujantau(n) \\ _Charles R Greathouse IV_, May 27 2016
%o A000594 (Sage) CuspForms( Gamma1(1), 12, prec=100).0; # _Michael Somos_, May 28 2013
%o A000594 (Sage) list(delta_qexp(100))[1:] # faster _Peter Luschny_, May 16 2016
%o A000594 (Ruby)
%o A000594 def s(n)
%o A000594 s = 0
%o A000594 (1..n).each{|i| s += i if n % i == 0}
%o A000594 s
%o A000594 end
%o A000594 def A000594(n)
%o A000594 ary = [1]
%o A000594 a = [0] + (1..n - 1).map{|i| s(i)}
%o A000594 (1..n - 1).each{|i| ary << (1..i).inject(0){|s, j| s - 24 * a[j] * ary[-j]} / i}
%o A000594 ary
%o A000594 end
%o A000594 p A000594(100) # _Seiichi Manyama_, Mar 26 2017
%o A000594 (Ruby)
%o A000594 def A000594(n)
%o A000594 ary = [0, 1]
%o A000594 (2..n).each{|i|
%o A000594 s, t, u = 0, 1, 0
%o A000594 (1..n).each{|j|
%o A000594 t += 9 * j
%o A000594 u += j
%o A000594 break if i <= u
%o A000594 s += (-1) ** (j % 2 + 1) * (2 * j + 1) * (i - t) * ary[-u]
%o A000594 }
%o A000594 ary << s / (i - 1)
%o A000594 }
%o A000594 ary[1..-1]
%o A000594 end
%o A000594 p A000594(100) # _Seiichi Manyama_, Nov 25 2017
%o A000594 (Python)
%o A000594 from sympy import divisor_sigma
%o A000594 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
%Y A000594 Cf. A076847 (tau(prime)), A278577 (prime powers), A037955, A027364, A037945, A037946, A037947, A008408 (Leech).
%Y A000594 For a(n) mod N for various values of N see A046694, A098108, A126812-...
%Y A000594 Cf. A006352, A099059, A099060, A262339, A292781.
%Y A000594 For primes p such that tau(p) == -1 (mod 23) see A106867.
%Y A000594 Cf. A010816, A004009. A013973.
%Y A000594 Cf. A126832(n) = a(n) mod 5.
%K A000594 sign,easy,core,mult,nice
%O A000594 1,2
%A A000594 _N. J. A. Sloane_