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

Author

Kevin O'Bryant, Feb 03 2024

Keywords

• nonn
• more

Comments

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

Examples

			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}.
		

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 7 of A365515.

Programs

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