0,1

For each nonnegative integer d, this sequence gives the largest value of m where the difference delta(m) = pi(gpf(m*(m+1))) - omega(m*(m+1)) equals d, given pi() is the prime counting function (A000720), gpf is the greatest prime factor (A006530), and omega counts distinct prime factors (A001221).

After m = a(d), all subsequent values of m have delta(m) > d. These represent critical transition points where consecutive smooth number pairs become increasingly rare.

The sequence illustrates double exponential growth driven by the Hardy-Ramanujan theorem: as omega(m) typically grows as log(log(m)), imposing a tight constraint between omega and the prime index forces m to grow as exp(exp(d)).

It is proved that each a(n) must exist (be finite) because the growth rate of pi(gpf(m*(m+1))) exceeds that of omega(m*(m+1)) asymptotically. However, no constructive bound is known for determining when the search can be terminated.

a(0) through a(2) computationally verified to m = 2*10^14 (A141399 verified a(0) to 2.29*10^25). We conjecture this is sufficient to justify the finality of a(0) and a(1), and likely the finality of a(2). See the distribution of delta values by running the Octave_Deltas.c program at the attached link.

Conjecture: All polynomial generator functions of order 2 or greater, G(n), with integer roots, and that are not perfect powers, are bounded in prime-complete numbers (A055932). Consequences: Since primorials are prime-complete by definition and verification shows no primorial products m*(m+1) for 715 <= m <= 633555, this suggests (714, 715) is the last pair of consecutive integers with primorial product. For Brocard's equation n! + 1 = m^2, since factorials are prime-complete and n! = (m-1)*(m+1), note that for odd m, (m-1)*(m+1) = 4*k*(k+1) where k = (m-1)/2, and delta(4*k*(k+1)) = delta(k*(k+1)) (multiplying by 4 just adds more powers of 2). Thus prime-complete (m-1)*(m+1) requires delta(k*(k+1)) = 0. Since a(0) = 633555 is the last k with delta(k*(k+1)) = 0, the bound m <= 1267111 (the final term of A194099) closes the search for solutions to Brocard's equation.

Ken Clements, Delta Distributions by Octave

a(0) = 633555 because when:

m = 633555 = 3^3 * 5 * 13 * 19^2 and

m+1 = 633556 = 2^2 * 7 * 11^3 * 17,

the combined product has gpf = 19 with pi(19) = 8 and omega = 8.

Thus d = 8 - 8 = 0.

This is the last m where delta achieves 0 (prime-complete).

a(1) = 80061344 because when:

m = 80061344 = 2^5 * 11^2 * 23 * 29 * 31 and

m+1 = 80061345 = 3^3 * 5 * 7^4 * 13 * 19,

the combined product has gpf = 31 with pi(31) = 11 and omega = 10.

Thus d = 11 - 10 = 1. This is the last m where the minimum d drops to 1.

a(2) = 1109496723125 because when:

m = 1109496723125 = 5^4 * 7 * 17 * 19^2 * 31^2 * 43 and

m+1 = 1109496723126 = 2 * 3 * 11^2 * 23 * 29^2 * 41^2 * 47,

the combined product highest prime index pi(47) = 15, with distinct primes = 13.

Thus d = 2, and this is the last m currently found where the minimum d drops to 2.

(Python, with numpy and numba)

""" Note: This code is an example. Because the search range is so large, it is only practical to run as multiple parallel threads, each taking a part of the search space. The parallelization is not shown here. """

from sympy import primerange

import numpy as np

from numba import njit

delta_limit, m_range = 3, 10_000_000_000_000

primes = np.array([0] + list(primerange(2, 100)))

@njit(cache=True)

def fast_po(m): # Return a fast Pidx and omega if they are small numbers

Pidx, omega = 0, 0

for i in range (1, 26):

p = primes[i]

if p > m:

return 99999, 0

div_hit = 1

while m % p == 0:

omega += div_hit

div_hit = 0

Pidx = i

m //= p

if m == 1:

return Pidx, omega

return 99999, 0

delta_min = [0] * delta_limit

Pidx_m, omega_m = 0, 0

for m in range(1, m_range+1):

Pidx_p1, omega_p1 = fast_po(m+1)

omega_product = omega_m + omega_p1

Pidx_product = max(Pidx_m, Pidx_p1)

Pidx_m, omega_m = Pidx_p1, omega_p1

delta = Pidx_product - omega_product

if delta < delta_limit:

delta_min[delta] = m

print(f"Final m = {m:, } and minimum deltas were at m = {delta_min}")

Cf. A391449, A252489, A059957, A055932, A141399, A002378, A391885, A391970.

Sequence in context: A183678 A267362 A392344 * A251564 A066590 A237909

Adjacent sequences: A391599 A391600 A391601 * A391603 A391604 A391605

nonn,hard,bref,more

Ken Clements, Dec 13 2025

approved