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.
0, 1, 7, 43, 154, 668, 2214, 6876, 16864, 41970, 94710, 202027, 429733, 889207, 1549511, 3238700, 5053317, 8502061, 15583775, 25070899, 40588284, 63604514
Offset: 1
Examples
a(5) != 50 because 50+1+1+1+1+0 = 43+7+1+1+1+1.
Links
- J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
- Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
- Melvyn 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, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.
Programs
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Python
def GreedyBh(h, seed, stopat): A = [set() for _ in range(h+1)] A[1] = set(seed) # A[i] will hold the i-fold sumset for j in range(2,h+1): # {2,...,h} for x in A[1]: A[j].update([x+y for y in A[j-1]]) w = max(A[1])+1 while w <= stopat: wgood = True for k in range(1,h): if wgood: for j in range(k+1,h+1): if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()): wgood = False if wgood: A[1].add(w) for k in range(2,h+1): # update A[k] for j in range(1,k): A[k].update([(k-j)*w + x for x in A[j]]) w += 1 return A[1] GreedyBh(6,[0],10000) -
Python
from itertools import count, islice, combinations_with_replacement def A365302_gen(): # generator of terms aset, alist = set(), [] for k in count(0): bset = set() for d in combinations_with_replacement(alist+[k],5): if (m:=sum(d)+k) in aset: break bset.add(m) else: yield k alist.append(k) aset |= bset A365302_list = list(islice(A365302_gen(),10)) # Chai Wah Wu, Sep 01 2023
Extensions
a(15)-a(19) from Chai Wah Wu, Sep 01 2023
a(20)-a(22) from Chai Wah Wu, Sep 09 2023
Comments