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A365302 a(n) is the smallest nonnegative integer such that the sum of any six ordered terms a(k), k<=n (repetitions allowed), is unique.

%I A365302 #44 Mar 28 2024 04:13:51
%S A365302 0,1,7,43,154,668,2214,6876,16864,41970,94710,202027,429733,889207,
%T A365302 1549511,3238700,5053317,8502061,15583775,25070899,40588284,63604514
%N A365302 a(n) is the smallest nonnegative integer such that the sum of any six ordered terms a(k), k<=n (repetitions allowed), is unique.
%C A365302 This is the greedy B_6 sequence.
%H A365302 J. Cilleruelo and J Jimenez-Urroz, <a href="https://doi.org/10.1112/S0025579300015758">B_h[g] sequences</a>, Mathematika (47) 2000, pp. 109-115.
%H A365302 Melvyn 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 A365302 Melvyn 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 A365302 Kevin O'Bryant, <a href="https://doi.org/10.37236/32">A complete annotated bibliography of work related to Sidon sequences</a>, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
%e A365302 a(5) != 50 because 50+1+1+1+1+0 = 43+7+1+1+1+1.
%o A365302 (Python)
%o A365302 def GreedyBh(h, seed, stopat):
%o A365302     A = [set() for _ in range(h+1)]
%o A365302     A[1] = set(seed)    # A[i] will hold the i-fold sumset
%o A365302     for j in range(2,h+1): # {2,...,h}
%o A365302         for x in A[1]:
%o A365302             A[j].update([x+y for y in A[j-1]])
%o A365302     w = max(A[1])+1
%o A365302     while w <= stopat:
%o A365302         wgood = True
%o A365302         for k in range(1,h):
%o A365302             if wgood:
%o A365302                 for j in range(k+1,h+1):
%o A365302                     if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
%o A365302                         wgood = False
%o A365302         if wgood:
%o A365302             A[1].add(w)
%o A365302             for k in range(2,h+1): # update A[k]
%o A365302                 for j in range(1,k):
%o A365302                     A[k].update([(k-j)*w + x for x in A[j]])
%o A365302         w += 1
%o A365302         return A[1]
%o A365302 GreedyBh(6,[0],10000)
%o A365302 (Python)
%o A365302 from itertools import count, islice, combinations_with_replacement
%o A365302 def A365302_gen(): # generator of terms
%o A365302     aset, alist = set(), []
%o A365302     for k in count(0):
%o A365302         bset = set()
%o A365302         for d in combinations_with_replacement(alist+[k],5):
%o A365302             if (m:=sum(d)+k) in aset:
%o A365302                 break
%o A365302             bset.add(m)
%o A365302         else:
%o A365302             yield k
%o A365302             alist.append(k)
%o A365302             aset |= bset
%o A365302 A365302_list = list(islice(A365302_gen(),10)) # _Chai Wah Wu_, Sep 01 2023
%Y A365302 Row 6 of A365515.
%Y A365302 Cf. A025582, A051912, A365300, A365301, A365303, A365304, A365305.
%K A365302 nonn,more
%O A365302 1,3
%A A365302 _Kevin O'Bryant_, Aug 31 2023
%E A365302 a(15)-a(19) from _Chai Wah Wu_, Sep 01 2023
%E A365302 a(20)-a(22) from _Chai Wah Wu_, Sep 09 2023