%I A000594 M5153 N2237
%S A000594 1,24,252,1472,4830,6048,16744,84480,113643,115920,534612,370944,
%T A000594 577738,401856,1217160,987136,6905934,2727432,10661420,7109760,4219488,
%U A000594 12830688,18643272,21288960,25499225,13865712,73279080,24647168
%V A000594 1,-24,252,-1472,4830,-6048,-16744,84480,-113643,-115920,534612,-370944,
%W A000594 -577738,401856,1217160,987136,-6905934,2727432,10661420,-7109760,-4219488,
%X A000594 -12830688,18643272,21288960,-25499225,13865712,-73279080,24647168
%N A000594 Ramanujan's tau function (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.
%C A000594 Number of partitions of n into an even number of distinct parts - partitions 
               of n into an odd number of distinct parts, with 24 types of each 
               part. - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
%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 (njas(AT)research.att.com), 
               Mar 25 2007
%D A000594 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000594 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000594 M. Boylan, Exceptional congruences for the coefficients of certain eta-product 
               newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A000594 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence 
               Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A000594 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. 
               Soc., 1988; p. 77, Eq. (32.2).
%D A000594 M. J. Hopkins, Algebraic toplogy and modular forms, Proc. Internat. Congress 
               Math., Beijing 2002, Vol. I, pp. 291-317.
%D A000594 Bruce Jordan and Blair Kelly (blair.kelly(AT)att.net), The vanishing 
               of the Ramanujan tau function, preprint, 2001.
%D A000594 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.
%D A000594 D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical 
               Journal, 1947, pp. 429-433.
%D A000594 D. H. Lehmer, Tables of Ramanujan's function tau(n), Math. Comp., 24 
               (1970), 495-496.
%D A000594 Yu. I. Manin, Mathematics and Physics, Birkhaeuser, Bosten, 1981.
%D A000594 H. McKean and V. Moll. Elliptic Curves, Camb. Univ. Press, p. 139.
%D A000594 S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi 
               elliptic functions, continued fractions and Schur functions, Ramanujan 
               J., 6 (2002), 7-149.
%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 S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of 
               Srinivasa Ramanujan, p. 153, Ed. G. H. Hardy et al., AMS Chelsea 
               2000.
%D A000594 S. Ramanujan, On Certain Arithmetical Functions. Ramanujan's Papers, 
               p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
%D A000594 J.-P. Serre, A course in Arithmetic, Springer-Verlag, 1973, see p. 98.
%D A000594 J.-P. Serre, Sur la lacunatit\'e des puissances de eta, Glasgow Math. 
               Journal, 27 (1985), 203-221.
%D A000594 J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, 
               Springer, see p. 482.
%D A000594 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.
%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 G. N. Watson, A table of Ramanujan's function tau(n), Proc. London Math. 
               Soc., 51 (1950), 1-13.
%D A000594 D. Zagier, Introduction to Modular Forms, Chapter 4 in M. Waldschmidt 
               et al., editors, From Number Theory to Physics, Springer-Verlag, 
               1992.
%H A000594 Simon Plouffe, <a href="b000594.txt">Table of n, a(n) for n = 1..16090</
               a>
%H A000594 S. R. Finch, <A HREF="http://algo.inria.fr/bsolve/">Modular forms on 
               SL_2(Z)</A>
%H A000594 B. C. Berndt and K. 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 B. C. Berndt and K. Ono, <a href="http://emis.dsd.sztaki.hu/journals/
               SLC/wpapers/s42berndt.html">Ramanujan's unpublished manuscript...</
               a>
%H A000594 F. Brunault, <a href="http://www.institut.math.jussieu.fr/~brunault/FonctionTau.pdf">
               La fonction Tau de Ramanujan</a>
%H A000594 N. A. Carella, <a href="http://arXiv.org/abs/math.NT/0512214">Note on 
               the Tau Function</a>
%H A000594 D. X. Charles, <a href="http://www.cs.wisc.edu/~cdx/CompTau.pdf">Computing 
               The Ramanujan Tau Function</a>
%H A000594 John Cremona, <a href="http://www.maths.nott.ac.uk/personal/jec">Home 
               page</a>
%H A000594 B. Edixhoven et al., <a href="http://arXiv.org/abs/math.NT/0605244">Computing 
               the coefficients of a modular form</a>
%H A000594 J. A. Ewell, <a href="http://www.ams.org/journal-getitem?pii=S0002-9939-99-05289-2#Abstract">
               Ramanujan's Tau Function</a>
%H A000594 J. A. Ewell, <a href="http://math.la.asu.edu/~rmmc/rmj/Vol28-2/EWE/EWE.html">
               Ramanujan's Tau Function</a>
%H A000594 M. Z. Garaev, V. C. Garcia and S. V. Konyagin, <a href="http://arXiv.org/
               abs/math.NT/0607169">Waring problem with the Ramanujan tau function</
               a>
%H A000594 J. L. Hafner and J. Stopple, The Ramanujan Journal 4(2) 2000, <a href="http:/
               /www.wkap.nl/oasis.htm/266553">A Heat Kernel Associated to Ramanujan's 
               Tau Function</a>
%H A000594 Jerry B. Keiper, <a href="http://mathsource.wri.com/MathSource22/Enhancements/
               NumberTheory/0200-978/Documentation.txt">Ramanujan's Tau-Dirichlet 
               Series</a>
%H A000594 F. Luca and I. E. Shparlinski, <a href="http://www.arXiv.org/abs/math.NT/
               0607591">Arithmetic properties of the Ramanujan function</a>
%H A000594 K. Matthews, <a href="http://www.numbertheory.org/php/tau.html">Computing 
               Ramanujan's tau function</a>
%H A000594 S. C. Milne, <a href="http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=26345">
               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 P. Moree, <a href="http://arXiv.org/pdf/math.NT/0201265">On some claims 
               in Ramanujan's 'unpublished' manuscript on the partition and tau 
               functions</a>
%H A000594 M. R. Murty, V. K. 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>
%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>
%H A000594 J. Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/ramanujantau/
               ramanujantau.htm">Ramanujan's Tau Function</a>
%H A000594 S. Plouffe, <a href="http://www.lacim.uqam.ca:16080/~plouffe/OEIS/b000594.txt">
               The first 225035 terms</a>
%H A000594 S. 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 J. P. Serre, <a href="http://public.csusm.edu/public/FranzL/publ/serre.pdf">
               An interpretation of some congruences concerning Ramanujan's tau 
               function</a>
%H A000594 J. P. Serre, <a href="http://citeseer.nj.nec.com/correct/477747">An interpretation 
               of some congruences concerning Ramanujan's Tau function</a>
%H A000594 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/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="http://arXiv.org/abs/math.CO/0207175">My Favorite 
               Integer Sequences</a>
%H A000594 D. A. Steffen, <a href="http://www.maths.mq.edu.au/~steffen/old/ramanujan/
               ramanujan.pdf">Les Coefficients de Fourier de la forme modulaire: 
               La fonction de Ramanujan tau(n)</a>
%H A000594 William Stein, <a href="http://modular.math.washington.edu/">Database</
               a>
%H A000594 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TauFunction.html">Link to a section of The World of Mathematics.</
               a>
%H A000594 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A000594 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A000594 G.f.: x Product_{k=1..infinity} (1 - x^k)^24.
%F A000594 |a(n)| = O(n^(11/2 + epsilon)), |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 formulae, 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)=uw(u+48v+4096w)-v^3. 
               - Michael Somos, Jul 19 2004
%e A000594 x Product (1 - x^k)^24 = x - 24*x^2 + 252*x^3 - 1472*x^4 + 4830*x^5 - 
               6048*x^6 - 16744*x^7 + 84480*x^8 - 113643*x^9 + ...
%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}] (from Dean Hickerson (dean.hickerson(AT)yahoo.com), Jan 
               03 2003)
%o A000594 (MAGMA) M12:=ModularForms(Gamma0(1),12); t1:=Basis(M12)[2]; PowerSeries(t1[1],
               100); Coefficients($1);
%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))
%Y A000594 Cf. A076847 (tau(p)), A037955, A027364, A037945, A037946, A037947, A008408 
               (Leech).
%Y A000594 For a(n) mod N for various values of N see A046694, A126811-...
%Y A000594 Sequence in context: A052652 A052732 A086603 this_sequence A022716 A051828 
               A076847
%Y A000594 Adjacent sequences: A000591 A000592 A000593 this_sequence A000595 A000596 
               A000597
%K A000594 sign,easy,core,mult,nice
%O A000594 1,2
%A A000594 N. J. A. Sloane (njas(AT)research.att.com).