OFFSET
0,1
COMMENTS
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.
From James Shakarji, Jun 15 2026: (Start)
Equivalently, delta(m) = d means m and m+1 are both P-smooth, where P = gpf(m*(m+1)), and m*(m+1) is divisible by all but exactly d of the primes <= P (the case d = 0 is m*(m+1) in A055932). Thus every m with a given value of delta is the smaller element of a pair of consecutive P-smooth integers. For each fixed prime set such pairs are finite and completely enumerable by Størmer's theorem in Lehmer's algorithmic form: with X = 2*m+1 and D the squarefree part of 4*m (m+1), they are the odd solutions of the Pell equation X^2 - D*Y^2 = 1 ranging over squarefree P-smooth D.
The sequence is not monotonic: a(4) = 20628591204480 > a(5) = 11259346386959. The per-delta maxima dip on the rising flank of the distribution of delta over consecutive smooth pairs, which peaks near delta = 8 and it is unconditional for gpf <= 89. (End)
LINKS
EXAMPLE
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 d 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 has gpf = 47 with pi(47) = 15 and omega = 13.
Thus d = 15 - 13 = 2. This is the last m currently found where the minimum d drops to 2.
a(3) = 1284729638049 because when:
m = 1284729638049 = 3^5 * 11 * 13^2 * 19 * 43 * 59^2 and
m+1 = 1284729638050 = 2 * 5^2 * 17 * 23^2 * 31 * 37 * 47 * 53,
the combined product has gpf = 59 with pi(59) = 17 and omega = 14.
Thus d = 17 - 14 = 3. This is the last m where the minimum d drops to 3 (verified for m <= 10^19).
a(4) = 20628591204480 because when:
m = 20628591204480 = 2^7 * 3 * 5 * 7 * 17^5 * 23 * 47 and
m+1 = 20628591204481 = 13^2 * 29 * 31 * 37^2 * 41^2 * 59,
the combined product has gpf = 59 with pi(59) = 17 and omega = 13.
Thus d = 17 - 13 = 4. This is the last m where the minimum d drops to 4 (verified for m <= 10^19).
a(5) = 11259346386959 because when:
m = 11259346386959 = 11^3 * 19 * 23^4 * 37 * 43 and
m+1 = 11259346386960 = 2^4 * 3^2 * 5 * 7 * 31^4 * 41 * 59,
the combined product has gpf = 59 with pi(59) = 17 and omega = 12.
Thus d = 17 - 12 = 5. This is the last m where the minimum d drops to 5 (verified for m <= 10^18).
a(6) = 119089041053696 because when:
m = 119089041053696 = 2^11 * 7 * 19 * 29 * 59^2 * 61 * 71 and
m+1 = 119089041053697 = 3^4 * 11^2 * 13 * 17 * 23^2 * 37 * 53^2,
the combined product has gpf = 71 with pi(71) = 20 and omega = 14.
Thus d = 20 - 14 = 6. This is the last m where the minimum d drops to 6 (verified for m <= 10^15).
PROG
(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}")
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Ken Clements, Dec 13 2025
EXTENSIONS
a(3)-a(6) from James Shakarji, Jun 15 2026
STATUS
approved