A005282 -id:A005282 - cp's OEIS frontend

A347570 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.

1, 1, 2, 1, 2, 3, 1, 2, 4, 4, 1, 2, 5, 8, 5, 1, 2, 6, 14, 13, 6, 1, 2, 7, 22, 33, 21, 7, 1, 2, 8, 32, 56, 72, 31, 8, 1, 2, 9, 44, 109, 154, 125, 45, 9, 1, 2, 10, 58, 155, 367, 369, 219, 66, 10, 1, 2, 11, 74, 257, 669, 927, 857, 376, 81, 11

Offset: 1

Peter Kagey, Sep 06 2021

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.

			Table begins:
n\k | 1  2   3   4    5     6     7      8
----+------------------------------------------
  1 | 1, 2,  3,  4,   5,    6,    7,     8, ...
  2 | 1, 2,  4,  8,  13,   21,   31,    45, ...
  3 | 1, 2,  5, 14,  33,   72,  125,   219, ...
  4 | 1, 2,  6, 22,  56,  154,  369,   857, ...
  5 | 1, 2,  7, 32, 109,  367,  927,  2287, ...
  6 | 1, 2,  8, 44, 155,  669, 2215,  6877, ...
  7 | 1, 2,  9, 58, 257, 1154, 4182, 14181, ...
  8 | 1, 2, 10, 74, 334, 1823, 8044, 28297, ...

Chai Wah Wu, Table of n, a(n) for n = 1..241
Eric Weisstein's World of Mathematics, B2 Sequence.

Cf. A000027 (n=1), A005282 (n=2), A096772 (n=3), A014206 (k=4), A370754 (k=5).

from itertools import count, islice, combinations_with_replacement
def A347570_gen(): # generator of terms
    asets, alists, klist = [set()], [[]], [1]
    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(1)
        asets.append(set())
        alists.append([])
A347570_list = list(islice(A347570_gen(),30)) # Chai Wah Wu, Sep 06 2023

A079848 Smallest primes such that a(j) - a(k) are all different.

Original entry on oeis.org

2, 3, 5, 11, 23, 37, 47, 97, 101, 149, 211, 233, 353, 383, 487, 641, 757, 797, 919, 1097, 1163, 1381, 1409, 1481, 1777, 1997, 2287, 2417, 2969, 3049, 3371, 3529, 3929, 4231, 4759, 5279, 5449, 5717, 5953, 6529, 6983, 7583, 8053, 8819, 9043, 10133, 10799

Offset: 1

Amarnath Murthy, Feb 18 2003

This is the slowest-growing prime B2 sequence. See A005282. - T. D. Noe, Mar 24 2007

T. D. Noe, Table of n, a(n) for n=1..4084 (terms less than 2*10^9)

Cf. A079849.

terms = 100; a = Table[0, {terms}]; s={}; k=0; A079848list = Reap[For[p=2, p < 10^5, p = NextPrime[p], j=1; While[j <= k && FreeQ[s, p-a[[j]]], j++]; If[j>k, For[j=1, j <= k, j++, s = Union[s, {p-a[[j]]}]]; k++; a[[k]] = p; Print[p]; Sow[p]; If[k == terms, Break[]]]]][[2, 1]]; (* Jean-François Alcover, Nov 02 2016, adapted from Max Alekseyev's PARI code *)

a=vector(100);s=Set();k=0;forprime(p=2,10^5,j=1;while(j<=k&&!setsearch(s,p-a[j]),j++);if(j>k, for(j=1,k,s=setunion(s,[p-a[j]]));k++;a[k]=p;print1(" ",p);if(k==100,break))) \\ Max Alekseyev, Feb 14 2005

from itertools import count, islice
from sympy import nextprime
def A079848_gen(): # generator of terms
    aset2, alist, k = set(), [], 0
    while (k:=nextprime(k)):
        bset2 = set()
        for a in alist:
            if (b:=k-a) in aset2:
                break
            bset2.add(b)
        else:
            yield k
            alist.append(k)
            aset2.update(bset2)
A079848_list = list(islice(A079848_gen(),30)) # Chai Wah Wu, Sep 11 2023

More terms from Max Alekseyev, Feb 14 2005

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

N. J. A. Sloane

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

R. K. Guy, Unsolved Problems in Number Theory, E25.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1000
R. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Cf. A054540, A004978, A247556.

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000

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

A080201 Numbers that do not occur as differences between terms of the Mian-Chowla variant A051788.

Original entry on oeis.org

49, 50, 71, 72, 76, 82, 90, 93, 95, 96, 119, 128, 139, 143, 152, 162, 172, 173, 180, 182, 185, 188

Offset: 1

Hugo Pfoertner, Feb 05 2003

Terms are only conjectures. Elements might be eliminated by higher terms of A051788, which was checked up to 4*10^6.

See under A005282.

Hugo Pfoertner, FORTRAN program to create the Mian-Chowla sequence

Cf. A051788, A005282, A080200.

A327762 a(n) = smallest positive number not already in the sequence such that all n(n+1)/2 numbers in the triangle of differences of the first n terms are distinct.

Original entry on oeis.org

1, 3, 9, 5, 12, 10, 23, 8, 22, 17, 42, 16, 43, 20, 38, 26, 45, 32, 65, 28, 64, 39, 76, 34, 81, 48, 98, 40, 92, 54, 109, 60, 116, 51, 114, 58, 117, 70, 136, 67, 135, 71, 145, 72, 147, 69, 146, 80, 164, 87, 166, 82, 170, 108, 198, 99

Offset: 1

N. J. A. Sloane, Sep 24 2019, revised Sep 25 2019

Inspired by A327743.
From Rémy Sigrist, Sep 25 2019: (Start)
The sequence is finite, with 56 terms.
Let b and c be the first and second differences of a, respectively, hence:
- b(55) = a(56) - a(55) = 99 - 198 = -99,
- b(56) = a(57) - a(56) = a(57) - 99,
- c(55) = b(56) - b(55) = a(57), a contradiction.
(End)
Since this definition leads to a finite sequence, it is natural to ask instead for the "Lexicographically earliest infinite sequence of distinct positive integers such that for every k >= 1, all the k(k+1)/2 numbers in the triangle of differences of the first k terms are distinct." This is A327460.
If only first differences are considered, one gets the classical Mian-Chowla sequence A005282. - M. F. Hasler, Oct 09 2019

			Difference triangle of the first k=8 terms of the sequence:
  1, 3, 9, 5, 12, 10, 23, 8, ...
  2, 6, -4, 7, -2, 13, -15, ...
  4, -10, 11, -9, 15, -28, ...
  -14, 21, -20, 24, -43, ...
  35, -41, 44, -67, ...
  -76, 85, -111, ...
  161, -196, ...
  -357, ...
All 8*9/2 = 36 numbers are distinct.

Cf. A327743, A327460.
For first differences see A327458; for the leading column of the difference triangle see A327459.
Cf. A005282.

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

A046185 A B2-sequence due to Rachel Lewis.

Original entry on oeis.org

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, 352, 415, 486, 540, 651, 706, 781, 864, 963, 1003, 1148, 1217, 1371, 1409, 1523, 1673, 1974, 2105, 2191, 2317, 2496, 2652, 2726, 2858, 3219, 3268, 3500, 3605, 3864, 3962, 4237

Offset: 0

Eric W. Weisstein

nonn

"The first 68 elements of the sequence are [given], then the greedy algorithm is used." For this B2-sequence: "The reciprocal sum is at least 2.16086," greater than that of the Mian-Chowla sequence (A005282), which is at most 2.158533. - Danny Rorabaugh (with quotes by Rachel Lewis), Sep 29 2015

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 163-166, with section 2.20.2, Mian-Chowla and B2-Sequences

Danny Rorabaugh, Table of n, a(n) for n = 0..848
Steven R. Finch, Erdos's Reciprocal Sum Constants [Broken link]
Steven R. Finch, Erdos's Reciprocal Sum Constants [From the Wayback machine]
Rachel Lewis, Mian-Chowla and B2 sequences, 1999. [Thanks to _Steven Finch_ for providing this document. Included with permission. - _N. J. A. Sloane_, Jan 02 2020]
Eric Weisstein's World of Mathematics, B2 Sequence

Cf. A005282.

For n>67, a(n) is the least number such that all pairwise differences of distinct elements of {a(0), ..., a(n)} are distinct. - Danny Rorabaugh, Sep 29 2015

a(34) corrected by Steven Finch in email with Danny Rorabaugh, Sep 29 2015

A058335 a(n) = least value such that sequence increases and pairwise differences are unique.

Original entry on oeis.org

1, 4, 5, 10, 12, 22, 35, 49, 64, 84, 100, 122, 141, 169, 225, 271, 295, 338, 399, 465, 547, 579, 670, 745, 816, 917, 993, 1033, 1172, 1258, 1401, 1533, 1644, 1789, 1878, 2106, 2257, 2419, 2571, 2724, 2942, 3006, 3148, 3308, 3475, 3719, 3991, 4272, 4428

Offset: 1

Gerald A. Mischke (Gerald.Mischke(AT)mutualofomaha.com), Dec 13 2000

nonn

A variation on A005282 (Mian-Chowla, where positive differences of pairs of elements are unique) by starting with a(1) =1, a(2) = 4.

Cf. A005282, A051788.

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

cp's OEIS Frontend

A347570 Table read by antidiagonals upward: the n-th row gives the lexicographically earliest infinite B_n sequence.

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A079848 Smallest primes such that a(j) - a(k) are all different.

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A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

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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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A080201 Numbers that do not occur as differences between terms of the Mian-Chowla variant A051788.

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A327762 a(n) = smallest positive number not already in the sequence such that all n(n+1)/2 numbers in the triangle of differences of the first n terms are 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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Haskell

Mathematica

Python

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A046185 A B2-sequence due to Rachel Lewis.

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