A025582 -id:A025582 - cp's OEIS frontend

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

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

A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 64, 81, 100, 169, 256, 289, 441, 484, 576, 625, 841, 1089, 1296, 1444, 1936, 2025, 2401, 2601, 3136, 4225, 4356, 4624, 5329, 5476, 5776, 6084, 7569, 9025, 10201, 11449, 11664, 12321, 12996, 13456, 14400, 16129, 17956, 20164, 22201

Offset: 1

Labos Elemer, Jul 02 2001

nonn

			36 is in the sequence since the pairwise sums of {1, 4, 9, 16, 25, 36} are all distinct: 2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 61, 72.
49 is not in the sequence since 1 + 49 = 25 + 25.

Klaus Brockhaus, Table of n, a(n) for n = 1..4944
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A000290, A011185, A010672, A025582, A005282, A062292, A133743, A133744.

from itertools import count, islice
def A062295_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in (n**2 for n in count(1)):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2 |= bset2
A062295_list = list(islice(A062295_gen(),30)) # Chai Wah Wu, Sep 05 2023

Edited, corrected and extended by Klaus Brockhaus, Sep 24 2007

A034757 a(1)=1, a(n) = smallest odd number such that all sums of pairs of (not necessarily distinct) terms in the sequence are distinct.

Original entry on oeis.org

1, 3, 7, 15, 25, 41, 61, 89, 131, 161, 193, 245, 295, 363, 407, 503, 579, 721, 801, 949, 1129, 1185, 1323, 1549, 1643, 1831, 1939, 2031, 2317, 2623, 2789, 3045, 3143, 3641, 3791, 4057, 4507, 4757, 5019, 5559, 5849, 6309, 6707, 7181, 7593

Offset: 1

Wouter Meeussen, Jun 01 2000

a(1) = 1, a(n) = least number such that every difference a(i)-a(j) is a distinct even number. - Amarnath Murthy, Apr 07 2004

			5 is not in the sequence since 5+1 is already obtainable from 3+3, 9 is excluded since 1, 3 and 7 are in the sequence and would collide with 1+9

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A025582, A051912, A055598.

Partial sums of A287178.

a034757 = (subtract 1) . (* 2) . a005282  -- Reinhard Zumkeller, Dec 18 2012

seq2={1, 3}; Do[le=Length[seq2]; t=Last[seq2]+2; While[Length[Expand[(Plus @@ (x^seq2) + x^t)^2]] < Pochhammer[3, le]/le!, t=t+2]; AppendTo[seq2, t], {20}]; Print@seq2

from itertools import count, islice
def A034757_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(1,2):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2.update(bset2)
A034757_list = list(islice(A034757_gen(),30)) # Chai Wah Wu, Sep 05 2023

a(n) = 2*A005282(n)-1. (David Wasserman)

An incorrect comment from Amarnath Murthy, also dated Apr 07 2004, has been deleted.

Offset fixed by Reinhard Zumkeller, Dec 18 2012

A062294 A B_2 sequence: a(n) is the smallest prime such that the pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 29, 47, 67, 83, 131, 163, 233, 307, 397, 443, 617, 727, 809, 941, 1063, 1217, 1399, 1487, 1579, 1931, 2029, 2137, 2237, 2659, 2777, 3187, 3659, 3917, 4549, 4877, 5197, 5471, 5981, 6733, 7207, 7349, 8039, 8291, 8543, 9283, 9689, 10037

Offset: 1

Labos Elemer, Jul 02 2001

nonn

Klaus Brockhaus, Table of n, a(n) for n = 1..3600
Eric Weisstein's World of Mathematics, B2-Sequence
Index entries for B_2 sequences

Cf. A011185, A010672, A025582, A005282, A061781, A061784, A062292, A079848, A133096.

from itertools import islice
from sympy import nextprime
def A062294_gen(): # generator of terms
    aset2, alist, k = set(), [], 0
    while (k:=nextprime(k)):
        bset2 = set()
        for a in alist:
            if (b:=a+k) in aset2:
                break
            bset2.add(b)
        else:
            yield k
            alist.append(k)
            aset2.update(bset2)
A062294_list = list(islice(A062294_gen(),30)) # Chai Wah Wu, Sep 11 2023

Edited, corrected and extended by Klaus Brockhaus, Sep 17 2007

A133097 a(n) = A005282(n) - A011185(n-1).

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 10, 15, 27, 28, 23, 28, 20, 30, 22, 40, 32, 45, 27, 62, 89, 62, 116, 167, 105, 118, 108, 51, 99, 151, 88, 137, 137, 265, 174, 195, 320, 321, 249, 283, 226, 281, 293, 394, 465, 369, 585, 565, 639, 404, 483, 221, 233, 428, 384, 370, 527, 431, 818

Offset: 1

Klaus Brockhaus, Sep 17 2007

sign

Also A025582(n) - A010672(n-1).
A005282 is the sequence of smallest numbers such that the pairwise sums of not necessarily distinct elements are all distinct, whereas A011185 is the sequence of smallest numbers such that the pairwise sums of distinct elements are all distinct.
Sequence has negative terms; the first one is a(65) = -130.

			a(6) = A005282(6) - A011185(6) = 21 - 13 = 8.

Klaus Brockhaus, Table of n, a(n) for n = 1..2400

Cf. A005282, A011185, A025582, A010672, A133096.

from itertools import count, islice
from collections import deque
def A133097_gen(): # generator of terms
    aset2, alist, bset2, blist, aqueue, bqueue = set(), [], set(), [], deque(), deque()
    for k in count(1):
        cset2 = {k<<1}
        if (k<<1) not in aset2:
            for a in alist:
                if (m:=a+k) in aset2:
                    break
                cset2.add(m)
            else:
                aqueue.append(k)
                alist.append(k)
                aset2.update(cset2)
        cset2 = set()
        for b in blist:
            if (m:=b+k) in bset2:
                break
            cset2.add(m)
        else:
            bqueue.append(k)
            blist.append(k)
            bset2.update(cset2)
        if len(aqueue) > 0 and len(bqueue) > 0:
            yield aqueue.popleft()-bqueue.popleft()
A133097_list = list(islice(A133097_gen(),30)) # Chai Wah Wu, Sep 11 2023

A062292 A B_2 sequence: a(n) is the smallest cube such that the pairwise sums of {a(1)...a(n)} are all distinct.

Original entry on oeis.org

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 35937, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336

Offset: 1

Labos Elemer, Jul 02 2001

nonn

A Mian-Chowla sequence consisting only of cubes.

			During recursive construction of this set, for n=1-50, the cubes of 12,18,24,32,34,36,48 are left out to keep all sums of distinct cubes distinct from each other.

Cf. A011185, A010672, A025582, A005282, A062294.

from itertools import count, islice
def A062292_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in (n**3 for n in count(1)):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2.update(bset2)
A062292_list = list(islice(A062292_gen(),30)) # Chai Wah Wu, Sep 05 2023

A080222 Record-setting differences between adjacent elements of the Mian-Chowla sequence A005282.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 14, 21, 26, 34, 48, 71, 74, 90, 113, 143, 153, 249, 270, 299, 346, 453, 535, 940, 1052, 1226, 1347, 1365, 2443, 2511, 4253, 4254, 6116, 7339, 8898, 13621, 15567, 17940, 21061, 21307, 25558, 35749, 39437, 46664, 62709

Offset: 1

Hugo Pfoertner, Feb 07 2003

nonn

			a(12)=71 because A005282(17)-A005282(16)=361-290=71 is greater than all previous differences. a(45)=A005282(619)-A005282(618)=3738616-3675907=62709

See under A005282.

Cf. A005282, A025582, A080200.

cp's OEIS Frontend

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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A062295 A B_2 sequence: a(n) is the smallest square such that pairwise sums of not necessarily distinct elements are all distinct.

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A034757 a(1)=1, a(n) = smallest odd number such that all sums of pairs of (not necessarily distinct) terms in the sequence are distinct.

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A062294 A B_2 sequence: a(n) is the smallest prime such that the pairwise sums of distinct elements are all distinct.

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