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a(n) is the smallest nonnegative integer such that the sum of any eight ordered terms a(k), k<=n (repetitions allowed), is unique.
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%I #37 Mar 07 2024 16:53:27

%S 0,1,9,73,333,1822,8043,28296,102042,338447,1054824,2569353,6237718,

%T 15947108,36179796

%N 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 This is the greedy B_8 sequence.

%H 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 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 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 a(4) != 70 because 70+1+1+0+0+0+0+0 = 9+9+9+9+9+9+9+0.

%o (Python)

%o def GreedyBh(h, seed, stopat):

%o A = [set() for _ in range(h+1)]

%o A[1] = set(seed) # A[i] will hold the i-fold sumset

%o for j in range(2,h+1): # {2,...,h}

%o for x in A[1]:

%o A[j].update([x+y for y in A[j-1]])

%o w = max(A[1])+1

%o while w <= stopat:

%o wgood = True

%o for k in range(1,h):

%o if wgood:

%o for j in range(k+1,h+1):

%o if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):

%o wgood = False

%o if wgood:

%o A[1].add(w)

%o for k in range(2,h+1): # update A[k]

%o for j in range(1,k):

%o A[k].update([(k-j)*w + x for x in A[j]])

%o w += 1

%o return A[1]

%o GreedyBh(8,[0],10000)

%o (Python)

%o from itertools import count, islice, combinations_with_replacement

%o def A365304_gen(): # generator of terms

%o aset, alist = set(), []

%o for k in count(0):

%o bset = set()

%o for d in combinations_with_replacement(alist+[k],7):

%o if (m:=sum(d)+k) in aset:

%o break

%o bset.add(m)

%o else:

%o yield k

%o alist.append(k)

%o aset |= bset

%o A365304_list = list(islice(A365304_gen(),10)) # _Chai Wah Wu_, Sep 01 2023

%Y Row 8 of A365515.

%Y Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365305.

%K nonn,more

%O 1,3

%A _Kevin O'Bryant_, Aug 31 2023

%E a(11)-a(15) from _Chai Wah Wu_, Sep 13 2023