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

1,1

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.

Links

Table of n, a(n) for n=1..47.

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.

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.

Example

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}.

Prog

(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

Crossrefs

Column 6 of A365515.

Cf. A369817.

Sequence in context: A154638 A054889 A056384 * A036683 A054444 A121332

Adjacent sequences: A369815 A369816 A369817 * A369819 A369820 A369821

Keyword

nonn

Author

Kevin O'Bryant, Feb 03 2024

Status

approved