A356468 Yu. V. Matiyasevich's Riemann Hypothesis test.

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

Table of n, a(n) for n=1..19.

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

Adjacent sequences: A356465 A356466 A356467 * A356469 A356470 A356471

Keyword

nonn

Author

Peter Luschny, Aug 08 2022

Status

approved