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A356468
Yu. V. Matiyasevich's Riemann Hypothesis test.
0
1, 10, 143, 1221, 21249, 274815, 5639631, 90945117, 1826620833, 38618333559, 1129082889375, 28218286333125, 915660945621585, 26435665650141135, 888640364800590255, 28827658089741286125, 1176745390297425986625, 43482016069074330150375, 1949108731388102309925375
OFFSET
1,2
COMMENTS
The sequence as defined by the Python program is infinite if and only if the Riemann Hypotheses is true, in which case the program never halts; otherwise it returns a negative number and halts.
LINKS
Yuri Matiyasevich, The Riemann Hypothesis in computer science, Theoretical Computer Science, Volume 807, 2020, Pages 257-265.
MATHEMATICA
Table[Factorial2[2 n + 3]/15 + PrimePi[n]^2 (Factorial2[2 n] - Factorial2[2 n - 2] Sum[(-1)^(k + 1)/k, {k, n - 1}] Floor[Log[2, LCM @@ Range[n]]])/2, {n, 20}] (* Eric W. Weisstein, Aug 25 2023 *)
PROG
(Python)
from math import gcd
def RiemannTest(stop):
d = m = p = 0
f0 = f1 = f3 = n = q = r = 1
b = True
# The condition "n <= stop" is added only
# to save the tester's resources.
while r >= 0 and n <= stop:
print(r, end=", ")
d *= 2 * n
d += -f1 if (b := not b) else f1
n += 1
g = gcd(n, q)
q = (n * q) // g
if g == 1: p += 1
m = 0
q2 = q
while q2 > 1:
q2 //= 2
m += d
f1 = 2 * f0
f0 *= 2 * n
f3 *= (2 * n + 3)
r = f3 - p * p * (m - f0)
if n < stop:
print("Bad luck, Bernhard!")
return r
RiemannTest(22)
CROSSREFS
Cf. A000720.
Sequence in context: A184710 A263055 A159327 * A276915 A284219 A180914
KEYWORD
nonn
AUTHOR
Peter Luschny, Aug 08 2022
STATUS
approved