Search: id:A000594
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%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, Table of n, a(n) for n = 1..16090
%H A000594 S. R. Finch, Modular forms on
SL_2(Z)
%H A000594 B. C. Berndt and K. Ono, Ramanujan's unpublished manuscript on the partition
and tau functions with proofs and commentary
%H A000594 B. C. Berndt and K. Ono, Ramanujan's unpublished manuscript...
%H A000594 F. Brunault,
La fonction Tau de Ramanujan
%H A000594 N. A. Carella, Note on
the Tau Function
%H A000594 D. X. Charles, Computing
The Ramanujan Tau Function
%H A000594 John Cremona, Home
page
%H A000594 B. Edixhoven et al., Computing
the coefficients of a modular form
%H A000594 J. A. Ewell,
Ramanujan's Tau Function
%H A000594 J. A. Ewell,
Ramanujan's Tau Function
%H A000594 M. Z. Garaev, V. C. Garcia and S. V. Konyagin, Waring problem with the Ramanujan tau function
%H A000594 J. L. Hafner and J. Stopple, The Ramanujan Journal 4(2) 2000, A Heat Kernel Associated to Ramanujan's
Tau Function
%H A000594 Jerry B. Keiper, Ramanujan's Tau-Dirichlet
Series
%H A000594 F. Luca and I. E. Shparlinski, Arithmetic properties of the Ramanujan function
%H A000594 K. Matthews, Computing
Ramanujan's tau function
%H A000594 S. 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.
%H A000594 P. Moree, On some claims
in Ramanujan's 'unpublished' manuscript on the partition and tau
functions
%H A000594 M. R. Murty, V. K. Murty and T. N. Shorey, Odd values of the Ramanujan tau-function
%H A000594 Oklahoma State Mathematics Department, Ramanujan
tau L-Function
%H A000594 J. Perry, Ramanujan's Tau Function
%H A000594 S. Plouffe,
The first 225035 terms
%H A000594 S. Ramanujan, Collected Papers, Table of tau(n);n=1
to 30
%H A000594 J. P. Serre,
An interpretation of some congruences concerning Ramanujan's tau
function
%H A000594 J. P. Serre, An interpretation
of some congruences concerning Ramanujan's Tau function
%H A000594 N. J. A. Sloane,
My favorite integer sequences, in Sequences and their Applications
(Proceedings of SETA '98).
%H A000594 N. J. A. Sloane, My Favorite
Integer Sequences
%H A000594 D. A. Steffen, Les Coefficients de Fourier de la forme modulaire:
La fonction de Ramanujan tau(n)
%H A000594 William Stein, Database
%H A000594 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
%H A000594 Index entries for "core" sequences
%H A000594 Index entries for expansions of Product_{k
>= 1} (1-x^k)^m
%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).
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