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A364087
Integers k for which A000594(k)^2 > 4 k^11, where A000594 is Ramanujan's tau function.
1
799, 1751, 2987, 3149, 3713, 4841, 5321, 6157, 6283, 6901, 7003, 7849, 8137, 8143, 8777, 8789, 9071, 9077, 10523, 10609, 10727, 10763, 10951, 11189, 12079, 12223, 12361, 12875, 12931, 13193, 13493, 13891, 14008, 14711, 14899, 14909, 15247, 15347, 15611, 15839, 15929, 16171, 16309, 16873
OFFSET
1,1
COMMENTS
Ramanujan conjectured that for prime values of k, A000594(k)^2 <= 4 k^11, where A000594 is Ramanujan's tau function. This followed from Deligne's proof of the Weil Conjectures. This sequence demonstrates the sparsity of small integers k such that the absolute value of tau(k) is greater than this bound.
LINKS
EXAMPLE
799 is a term: tau(799) = -18558651348375264 while 2*799^(11/2) = 18409300288204184.
PROG
(Julia)
using Nemo
function bigger_than_prime_bound(len)
R, z = PolynomialRing(ZZ, "z")
exp = eta_qexp(24, len, z)
t = BigInt.([coeff(exp, j) for j in 0:len - 1])
n_vals = []
for i in eachindex(t)
if abs(t[i]) > 2 * i ^ (11/2)
push!(n_vals, i)
end
end
n_vals
end
(PARI) isok(k) = ramanujantau(k)^2 > 4*k^11; \\ Michel Marcus, Jul 05 2023
(Python)
from itertools import count, islice
from sympy import divisor_sigma
def A364087_gen(startvalue=1): # generator of terms >= startvalue
return (n for n in count(max(startvalue, 1)) if (n**11<<2) < (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))))**2)
A364087_list = list(islice(A364087_gen(), 10)) # Chai Wah Wu, Aug 24 2023
CROSSREFS
Cf. A000594.
Sequence in context: A045261 A101039 A121642 * A035886 A283899 A043404
KEYWORD
nonn
AUTHOR
Jack Gilchrist, Jul 04 2023
STATUS
approved