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

1,3

Comments

This is the greedy B_6 sequence.

Links

Table of n, a(n) for n=1..22.

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.

Example

a(5) != 50 because 50+1+1+1+1+0 = 43+7+1+1+1+1.

Prog

(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

Crossrefs

Row 6 of A365515.

Cf. A025582, A051912, A365300, A365301, A365303, A365304, A365305.

Sequence in context: A183537 A114352 A201707 * A255314 A166813 A112563

Adjacent sequences: A365299 A365300 A365301 * A365303 A365304 A365305

Keyword

nonn,more

Author

Kevin O'Bryant, Aug 31 2023

Extensions

a(15)-a(19) from Chai Wah Wu, Sep 01 2023

a(20)-a(22) from Chai Wah Wu, Sep 09 2023

Status

approved