A369818 The sixth term of the greedy B_n set of natural numbers.

5, 20, 71, 153, 366, 668, 1153, 1822, 3119, 4448, 6348, 8559, 11565, 14976, 21023, 26220, 33066, 40306, 49601, 59354, 76031, 89248, 106008, 122909, 143989, 165196, 200759, 227660, 261030, 293736, 333825, 373110, 438191, 485952, 544356, 600523, 668573, 734072, 841679, 918988, 1012578, 1101374, 1208065, 1309426, 1474943, 1592000, 1732656

Offset: 1

Author

Kevin O'Bryant, Feb 03 2024

Keywords

• nonn

Comments

{0, 1, n+1, n^2+n+1, A369817(n), a(n)} is the lexicographically first set of 6 nonnegative integers with the property that the sum of any n nondecreasing terms (repetitions allowed) is unique.

Examples

			a(2) = 20, as all 21 nonincreasing sums from {0,1,3,7,12,20}, namely 0+0 < 0+1 < 1+1 < 0+3 < 1+3 < 3+3 < 0+7 < 1+7 < 3+7 < 0+12 < 1+12 < 7+7 < 3+12 < 7+12 < 0+20 < 1+20 < 3+20 < 12+12 < 7+20 < 12+20 < 20+20, are distinct, and all other 6-element sets of nonnegative integers with this property are lexicographically after {0,1,3,7,12,20}.
		

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.

Crossrefs

Column 6 of A365515.

Cf. A369817.

Programs

Python

from itertools import count, combinations_with_replacement
def A369818(n):
    alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1)]
    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(max(alist[-1]+1,(n**3>>1)*(1+(n>>2)))):
        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

Formula

Conjectured that a(6n+i) is a quartic polynomial sequence with lead term (1/3)n^4 for each i in {1,2,3,5,6,10} in arxiv:2312.10910.

Proved that (1/8)*n^4 + (1/2)*n^3 <= a(n) <= 0.406671*n^4 + O(n^3) in arxiv:2312.10910.