A365301 -id:A365301 - cp's OEIS frontend

A365515 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence starting from 0.

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 7, 4, 0, 1, 5, 13, 12, 5, 0, 1, 6, 21, 32, 20, 6, 0, 1, 7, 31, 55, 71, 30, 7, 0, 1, 8, 43, 108, 153, 124, 44, 8, 0, 1, 9, 57, 154, 366, 368, 218, 65, 9, 0, 1, 10, 73, 256, 668, 926, 856, 375, 80, 10, 0, 1, 11, 91, 333, 1153, 2214, 2286, 1424, 572, 96, 11

Offset: 1

Chai Wah Wu, Sep 07 2023

A B_n sequence is a sequence such that all sums a(x_1) + a(x_2) + ... + a(x_n) are distinct for 1 <= x_1 <= x_2 <= ... <= x_n. Analogous to A347570 except that here the B_n sequences start from a(1) = 0.

			Table begins:
n\k | 1  2   3   4    5     6      7      8       9
----+---------------------------------------------------
  1 | 0, 1,  2,  3,   4,    5,     6,     7,      8, ...
  2 | 0, 1,  3,  7,  12,   20,    30,    44,     65, ...
  3 | 0, 1,  4, 13,  32,   71,   124,   218,    375, ...
  4 | 0, 1,  5, 21,  55,  153,   368,   856,   1424, ...
  5 | 0, 1,  6, 31, 108,  366,   926,  2286,   5733, ...
  6 | 0, 1,  7, 43, 154,  668,  2214,  6876,  16864, ...
  7 | 0, 1,  8, 57, 256, 1153,  4181, 14180,  47381, ...
  8 | 0, 1,  9, 73, 333, 1822,  8043, 28296, 102042, ...
  9 | 0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, ...

Chai Wah Wu, Table of n, a(n) for n = 1..242
Eric Weisstein's World of Mathematics, B2 Sequence.

Cf. A001477 (n=1), A025582 (n=2), A051912 (n=3), A365300 (n=4), A365301 (n=5), A365302 (n=6), A365303 (n=7), A365304 (n=8), A365305 (n=9), A002061 (k=4), A369817 (k=5), A369818 (k=6), A369819 (k=7), A347570.

from itertools import count, islice, combinations_with_replacement
def A365515_gen(): # generator of terms
    asets, alists, klist = [set()], [[]], [0]
    while True:
        for i in range(len(klist)-1,-1,-1):
            kstart, alist, aset = klist[i], alists[i], asets[i]
            for k in count(kstart):
                bset = set()
                for d in combinations_with_replacement(alist+[k],i):
                    if (m:=sum(d)+k) in aset:
                        break
                    bset.add(m)
                else:
                    yield k
                    alists[i].append(k)
                    klist[i] = k+1
                    asets[i].update(bset)
                    break
        klist.append(0)
        asets.append(set())
        alists.append([])
A365515_list = list(islice(A365515_gen(),30))

a(n) = A347570(n)-1.

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.

Original entry on oeis.org

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

Kevin O'Bryant, Aug 31 2023

nonn

This is the greedy B_4 sequence.

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

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.

Row 4 of A365515.

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

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)

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

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

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.

Original entry on oeis.org

0, 1, 7, 43, 154, 668, 2214, 6876, 16864, 41970, 94710, 202027, 429733, 889207, 1549511, 3238700, 5053317, 8502061, 15583775, 25070899, 40588284, 63604514

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_6 sequence.

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

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.

Row 6 of A365515.

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

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)

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

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

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

A365303 a(n) is the smallest nonnegative integer such that the sum of any seven ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 8, 57, 256, 1153, 4181, 14180, 47381, 115267, 307214, 737909, 1682367, 3850940, 8557010, 18311575, 37925058, 61662056

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_7 sequence.

			a(3) != 7 because 7+0+0+0+0+0+0 = 1+1+1+1+1+1+1.

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.

Row 7 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365304, A365305.

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(7,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365303_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],6):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365303_list = list(islice(A365303_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(13)-a(18) from Chai Wah Wu, Sep 13 2023

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.

Original entry on oeis.org

0, 1, 9, 73, 333, 1822, 8043, 28296, 102042, 338447, 1054824, 2569353, 6237718, 15947108, 36179796

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_8 sequence.

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

J. Cilleruelo and J Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
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.

Row 8 of A365515.

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

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(8,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365304_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],7):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365304_list = list(islice(A365304_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(11)-a(15) from Chai Wah Wu, Sep 13 2023

A365305 a(n) is the smallest nonnegative integer such that the sum of any nine ordered terms a(k), k<=n (repetitions allowed), is unique.

Original entry on oeis.org

0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, 742330, 2464208, 7616100, 19241477, 56562573

Offset: 1

Kevin O'Bryant, Aug 31 2023

This is the greedy B_9 sequence.

			a(4) != 72 because 72+1+1+1+1+1+1+1+1+0 = 10+10+10+10+10+10+10+10+0.

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.

Row 9 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365304.

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(9,[0],10000)

from itertools import count, islice, combinations_with_replacement
def A365305_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],8):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365305_list = list(islice(A365305_gen(),10)) # Chai Wah Wu, Sep 01 2023

a(11)-a(14) from Chai Wah Wu, Sep 13 2023

cp's OEIS Frontend

A365515 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence starting from 0.

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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.

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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.

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A365303 a(n) is the smallest nonnegative integer such that the sum of any seven ordered terms a(k), k<=n (repetitions allowed), is unique.

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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.

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A365305 a(n) is the smallest nonnegative integer such that the sum of any nine ordered terms a(k), k<=n (repetitions allowed), is unique.

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