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A076847
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Ramanujan function tau(p) as p runs through the primes.
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10
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-24, 252, 4830, -16744, 534612, -577738, -6905934, 10661420, 18643272, 128406630, -52843168, -182213314, 308120442, -17125708, 2687348496, -1596055698, -5189203740, 6956478662, -15481826884, 9791485272, 1463791322, 38116845680, -29335099668, -24992917110
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OFFSET
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1,1
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COMMENTS
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This sequence determines all values of Ramanujan's tau function A000594 due to alpha-multiplicativity with alpha(x) = x^11 (the weight of the modular cusp form eta^{24}(z) with the Dedekind eta function is k = 12). See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 114, eq. (3) for the tau function. This implies multiplicativity of tau with tau(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. See the Apostol Exercise 6 on p. 139.
Note that the product representation of the Dirichlet series Sum_{n >=1} tau(n)/Sum_{n >= 1} tau(n)/n^s = Prod_{n >= 1} 1/(1 - a(n)/prime(n)^s + prime(n)^(11) / prime(n)^(2*s)) (see the Mordell reference, eq. (2)) leads also to this formula for tau(p^k) for primes p after expanding the factors of the product and collecting powers of 1/p^(k*s). If one insists on convergence of the product one can use s >= 7, if one uses Ramanujan's 1916 conjecture (proved by P. Deligne 1974) |tau(p)| <= 2*p^(11/2), i.e., |a(n)| <= 2*sqrt(prime(n)^11).
(End)
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REFERENCES
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Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 114, 138-139.
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LINKS
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FORMULA
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a(n)*a(m) = A000594(prime(n)*prime(m)) for n != m (from the tau multiplicativity). - Wolfdieter Lang, May 15 2016
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EXAMPLE
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84480 = A000594(2^3) = sqrt(2^(11))^3*S(3, -24/sqrt(2^(11))) = (-24)*((-24)^2 -2*2^11) = 84480. - Wolfdieter Lang, May 15 2016
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MATHEMATICA
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PROG
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(PARI) taup(p)=(65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756
(PARI) H(n)=sumdiv(core(n, 1)[2], d, my(D=-n/d^2); if(D%4<2, qfbclassno(D)/max(1, D+6)))
taup(p)=my(x='x, P=x^5-9*p*x^4+28*p^2*x^3-35*p^3*x^2+15*p^4*x-p^5); p^5*H(4*p)/2-1-sum(t=1, sqrtint(4*p), subst(P, x, t^2)*H(4*p-t^2))
(Perl) use ntheory ":all"; forprimes { say ramanujan_tau($_) } 100 # Dana Jacobsen, Sep 05 2015
(Sage)
[p for (n, p) in enumerate(list(delta_qexp(100))) if is_prime(n)] # Peter Luschny, May 16 2016
(Python)
from sympy import prime, divisor_sigma
def A076847(n): return -24 if n == 1 else (q:=(p:=prime(n))**4)*(p+1)-24*(sum((i*(i*(i*(70*i - 140*p) + 90*p**2) - 20*p**3) + q)*divisor_sigma(i)*divisor_sigma(p-i) for i in range(1, p+1>>1))) # Chai Wah Wu, Nov 09 2022
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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