A369819 The seventh term of the greedy B_n set of natural numbers.

6, 30, 124, 368, 926, 2214, 4181, 8043, 13818, 23614, 34825, 54011, 84026, 109870, 156474, 217790, 304910, 376260, 510220, 667130, 794873, 1008048, 1302947, 1629264, 1916949, 2361150, 2859694, 3467661, 3989744, 4779270, 5479857, 6449983, 7575912

Offset

1,1

Comments

Proved in arXiv:2312.10910 that a(n) <= 0.382978*n^5 + O(n^4).

Links

Table of n, a(n) for n=1..33.

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

Column 7 of A365515.

Sequence in context: A215225 A056385 A266945 * A073970 A074009 A248328

Adjacent sequences: A369816 A369817 A369818 * A369820 A369821 A369822

Keyword

nonn,more

Author

Kevin O'Bryant, Feb 03 2024

Status

approved