OFFSET
1,3
COMMENTS
Erdős and Graham believed that a(n) = (1+o(1))*n, but Liu and Sawhney have proved that a(n) = (1 - 1/e + o(1))*n.
LINKS
Thomas Bloom, Problem 300, Erdős Problems.
P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).
Erdős problems database contributors, Issue #152 linking Erdős problems to the OEIS.
Yang P. Liu and Mehtaab Sawhney, On further questions regarding unit fractions, arXiv:2404.07113 [math.NT], 2024.
Husnain Raza, Python program
Peter J. Taylor, Python program
PROG
(Python)
from math import prod
from itertools import combinations
def A390393(n):
s, t = [], set()
for l in range(1, n+1):
for p in combinations(range(1, n+1), r=l):
m = prod(p)
if sum(m//d for d in p)==m:
s.append(c:=set(p))
t |= c
l = len(t)
for i in range(l, -1, -1):
for d in combinations(t, i):
if not any(x.issubset(d) for x in s):
return n-l+i # Chai Wah Wu, Nov 23 2025
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Husnain Raza, Nov 04 2025
EXTENSIONS
a(30)-a(50) from Peter J. Taylor, Nov 08 2025
a(51)-a(59) from Chai Wah Wu, Nov 23 2025
STATUS
approved