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%I A369818 #26 Feb 29 2024 06:25:46
%S A369818 5,20,71,153,366,668,1153,1822,3119,4448,6348,8559,11565,14976,21023,
%T A369818 26220,33066,40306,49601,59354,76031,89248,106008,122909,143989,
%U A369818 165196,200759,227660,261030,293736,333825,373110,438191,485952,544356,600523,668573,734072,841679,918988,1012578,1101374,1208065,1309426,1474943,1592000,1732656
%N A369818 The sixth term of the greedy B_n set of natural numbers.
%C A369818 {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.
%H A369818 M. B. Nathanson, <a href="https://arxiv.org/abs/2310.14426">The third positive element in the greedy B_h-set</a>, arXiv:2310.14426 [math.NT], 2023.
%H A369818 M. B. Nathanson and Kevin O'Bryant, <a href="https://arxiv.org/abs/2311.14021">The fourth positive element in the greedy B_h-set</a>, arXiv:2311.14021 [math.NT], 2023.
%H A369818 Kevin O'Bryant, <a href="https://arxiv.org/abs/2312.10910">B_h-sets and Rigidity</a>, arXiv:2312.10910 [math.NT], 2023.
%F A369818 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.
%F A369818 Proved that (1/8)*n^4 + (1/2)*n^3 <= a(n) <= 0.406671*n^4 + O(n^3) in arxiv:2312.10910.
%e A369818 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}.
%o A369818 (Python)
%o A369818 from itertools import count, combinations_with_replacement
%o A369818 def A369818(n):
%o A369818     alist = [0,1,n+1,n*(n+1)+1,(n+3>>1)*n**2+(3*n+2>>1)]
%o A369818     aset = set(sum(d) for d in combinations_with_replacement(alist,n))
%o A369818     blist = []
%o A369818     for i in range(n):
%o A369818         blist.append(set(sum(d) for d in combinations_with_replacement(alist,i)))
%o A369818     for k in count(max(alist[-1]+1,(n**3>>1)*(1+(n>>2)))):
%o A369818         for i in range(n):
%o A369818             if any((n-i)*k+d in aset for d in blist[i]):
%o A369818                 break
%o A369818         else:
%o A369818             return k # _Chai Wah Wu_, Feb 28 2024
%Y A369818 Column 6 of A365515.
%Y A369818 Cf. A369817.
%K A369818 nonn
%O A369818 1,1
%A A369818 _Kevin O'Bryant_, Feb 03 2024

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