OFFSET
1,1
COMMENTS
Proved in arXiv:2312.10910 that a(n) <= 0.382978*n^5 + O(n^4).
LINKS
M. B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
M. B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
Kevin O'Bryant, B_h-sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
EXAMPLE
a(2) = 30, as all 28 nonincreasing sums from {0,1,3,7,12,20,30}, namely 0+0 < 0+1 < 1+1 < ... < 7+20 < 0+30 < 1+30 < 12+20 <3+30 < 7+30 < 20+20 < 12+30 < 20+30 < 30+30, are distinct, and all other 7-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20,30}.
PROG
(Python)
# uses Python code from A369818
from itertools import count, combinations_with_replacement
def A369819(n):
alist = [0, 1, n+1, n*(n+1)+1, (n+3>>1)*n**2+(3*n+2>>1), A369818(n)]
aset = set(sum(d) for d in combinations_with_replacement(alist, n))
blist = []
for i in range(n):
blist.append(set(sum(d) for d in combinations_with_replacement(alist, i)))
for k in count(alist[-1]+1):
for i in range(n):
if any((n-i)*k+d in aset for d in blist[i]):
break
else:
return k # Chai Wah Wu, Feb 28 2024
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Kevin O'Bryant, Feb 03 2024
STATUS
approved