A365300 a(n) is the smallest nonnegative integer such that the sum of any four ordered terms a(k), k<=n (repetitions allowed), is unique.

0, 1, 5, 21, 55, 153, 368, 856, 1424, 2603, 4967, 8194, 13663, 22432, 28169, 47688, 65545, 96615, 146248, 202507, 266267, 364834, 450308, 585328, 773000, 986339, 1162748, 1472659, 1993180, 2275962, 3012656, 3552307, 4590959, 5404183, 6601787, 7893270, 9340877

Offset: 1

Author

Kevin O'Bryant, Aug 31 2023

Keywords

• nonn

Comments

This is the greedy B_4 sequence.

Examples

			a(4) != 12 because 12+1+1+1 = 5+5+5+0.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..50
• 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.

Crossrefs

Row 4 of A365515.

Cf. A025582, A051912, A365301, A365302, A365303, A365304, A365305.

Programs

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(4,[0],10000)

Python

from itertools import count, islice, combinations_with_replacement
def A365300_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],3):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365300_list = list(islice(A365300_gen(),20)) # Chai Wah Wu, Sep 01 2023

Extensions

a(27)-a(37) from Chai Wah Wu, Sep 01 2023