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A365304 a(n) is the smallest nonnegative integer such that the sum of any eight ordered terms a(k), k<=n (repetitions allowed), is unique.

%I A365304 #37 Mar 07 2024 16:53:27
%S A365304 0,1,9,73,333,1822,8043,28296,102042,338447,1054824,2569353,6237718,
%T A365304 15947108,36179796
%N A365304 a(n) is the smallest nonnegative integer such that the sum of any eight ordered terms a(k), k<=n (repetitions allowed), is unique.
%C A365304 This is the greedy B_8 sequence.
%H A365304 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 A365304 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 A365304 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 A365304 a(4) != 70 because 70+1+1+0+0+0+0+0 = 9+9+9+9+9+9+9+0.
%o A365304 (Python)
%o A365304 def GreedyBh(h, seed, stopat):
%o A365304     A = [set() for _ in range(h+1)]
%o A365304     A[1] = set(seed)    # A[i] will hold the i-fold sumset
%o A365304     for j in range(2,h+1): # {2,...,h}
%o A365304         for x in A[1]:
%o A365304             A[j].update([x+y for y in A[j-1]])
%o A365304     w = max(A[1])+1
%o A365304     while w <= stopat:
%o A365304         wgood = True
%o A365304         for k in range(1,h):
%o A365304             if wgood:
%o A365304                 for j in range(k+1,h+1):
%o A365304                     if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
%o A365304                         wgood = False
%o A365304         if wgood:
%o A365304             A[1].add(w)
%o A365304             for k in range(2,h+1): # update A[k]
%o A365304                 for j in range(1,k):
%o A365304                     A[k].update([(k-j)*w + x for x in A[j]])
%o A365304         w += 1
%o A365304         return A[1]
%o A365304 GreedyBh(8,[0],10000)
%o A365304 (Python)
%o A365304 from itertools import count, islice, combinations_with_replacement
%o A365304 def A365304_gen(): # generator of terms
%o A365304     aset, alist = set(), []
%o A365304     for k in count(0):
%o A365304         bset = set()
%o A365304         for d in combinations_with_replacement(alist+[k],7):
%o A365304             if (m:=sum(d)+k) in aset:
%o A365304                 break
%o A365304             bset.add(m)
%o A365304         else:
%o A365304             yield k
%o A365304             alist.append(k)
%o A365304             aset |= bset
%o A365304 A365304_list = list(islice(A365304_gen(),10)) # _Chai Wah Wu_, Sep 01 2023
%Y A365304 Row 8 of A365515.
%Y A365304 Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365305.
%K A365304 nonn,more
%O A365304 1,3
%A A365304 _Kevin O'Bryant_, Aug 31 2023
%E A365304 a(11)-a(15) from _Chai Wah Wu_, Sep 13 2023