A080200 -id:A080200 - cp's OEIS frontend

A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, 565, 593, 662, 775, 822, 916, 970, 1016, 1159, 1312, 1395, 1523, 1572, 1821, 1896, 2029, 2254, 2379, 2510, 2780, 2925, 3155, 3354, 3591, 3797, 3998, 4297, 4433, 4779, 4851

Offset: 1

N. J. A. Sloane and Simon Plouffe

An alternative definition is to start with 1 and then continue with the least number such that all pairwise differences of distinct elements are all distinct. - Jens Voß, Feb 04 2003. [However, compare A003022 and A227590. - N. J. A. Sloane, Apr 08 2016]
Rachel Lewis points out [see link] that S, the sum of the reciprocals of this sequence, satisfies 2.158435 <= S <= 2.158677. Similarly, the sum of the squares of reciprocals of this sequence converges to approximately 1.33853369 and the sum of the cube of reciprocals of this sequence converges to approximately 1.14319352. - Jonathan Vos Post, Nov 21 2004; comment changed by N. J. A. Sloane, Jan 02 2020
Let S denote the reciprocal sum of a(n). Then 2.158452685 <= S <= 2.158532684. - Raffaele Salvia, Jul 19 2014
From Thomas Ordowski, Sep 19 2014: (Start)
Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).
Conjecture: a(n) ~ n^3 / log(n)^2. (End)

			The second term is 2 because the 3 pairwise sums 1+1=2, 1+2=3, 2+2=4 are all distinct.
The third term cannot be 3 because 1+3 = 2+2. But it can be 4, since 1+4=5, 2+4=6, 4+4=8 are distinct and distinct from the earlier sums 1+1=2, 1+2=3, 2+2=4.

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.20.2.
R. K. Guy, Unsolved Problems in Number Theory, E28.
A. M. Mian and S. D. Chowla, On the B_2-sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 3-4.
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..5818 (terms less than 2*10^9)
Thomas Bloom, Problem 340, Erdős Problems.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
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]
Kevin O'Bryant, B_h-Sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
Raffaele Salvia, Table of n, a(n) for n=1...25000
R. Salvia, A New Lower Bound for the Distinct Distance Constant, J. Int. Seq. 18 (2015) # 15.4.8.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Eric Weisstein's World of Mathematics, B2 Sequence.
Eric Weisstein's World of Mathematics, Chowla Sequence.
Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.
Index entries for B_2 sequences.

Row 2 of A347570.
Cf. A051788, A080200 (for differences between terms).
Different from A046185. Cf. A011185.
See also A003022, A227590.
A259964 has a greater sum of reciprocals.
Cf. A002858.

import Data.Set (Set, empty, insert, member)
a005282 n = a005282_list !! (n-1)
a005282_list = sMianChowla [] 1 empty where
   sMianChowla :: [Integer] -> Integer -> Set Integer -> [Integer]
   sMianChowla sums z s | s' == empty = sMianChowla sums (z+1) s
                        | otherwise   = z : sMianChowla (z:sums) (z+1) s
      where s' = try (z:sums) s
            try :: [Integer] -> Set Integer -> Set Integer
            try []     s                      = s
            try (x:sums) s | (z+x) `member` s = empty
                           | otherwise        = try sums $ insert (z+x) s
-- Reinhard Zumkeller, Mar 02 2011

a[1]:= 1: P:= {2}: A:= {1}:
for n from 2 to 100 do
  for t from a[n-1]+1 do
    Pt:= map(`+`,A union {t},t);
    if Pt intersect P = {} then break fi
  od:
  a[n]:= t;
  A:= A union {t};
  P:= P union Pt;
od:
seq(a[n],n=1..100); # Robert Israel, Sep 21 2014

t = {1}; sms = {2}; k = 1; Do[k++; While[Intersection[sms, k + t] != {}, k++]; sms = Join[sms, t + k, {2 k}]; AppendTo[t, k], {49}]; t (* T. D. Noe, Mar 02 2011 *)

A005282_vec(N, A=[1], U=[0], D(A, n=#A)=vector(n-1, k, A[n]-A[n-k]))={ while(#A2 && U=U[k-1..-1]);A} \\ M. F. Hasler, Oct 09 2019

aupto(L)= my(S=vector(L), A=[1]); for(i=2, L, for(j=1, #A, if(S[i-A[j]], next(2))); for(j=1, #A, S[i-A[j]]=1); A=concat(A, i)); A \\ Ruud H.G. van Tol, Jun 30 2025

from itertools import count, islice
def A005282_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k 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
A005282_list = list(islice(A005282_gen(),30)) # Chai Wah Wu, Sep 05 2023

a(n) = A025582(n) + 1.

a(n) = (A034757(n)+1)/2.

Examples added by N. J. A. Sloane, Jun 01 2008

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.

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.

A080932 Non-occurring pairwise differences between the elements of the Mian-Chowla sequence variant A058335.

Original entry on oeis.org

26, 33, 50, 55, 68, 93, 94, 98, 107, 108, 109, 115, 122, 138, 144, 150, 155, 163, 178, 181, 182, 183, 185, 186, 193, 196, 200, 202, 204, 208, 210, 212, 223, 227, 232, 235, 239, 242, 245, 250, 253, 257, 263, 264, 268

Offset: 1

Hugo Pfoertner, Feb 24 2003

See under A005282.

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

Cf. A058335, A005282, A080200, A080201, A080931.

A247556 Exact differential base (a B_2 sequence) constructed as follows: Start with a(0)=0. For n>=1, let S be the set of all differences a(j)-a(i) for 0 <= i < j <= n-1, and let d be the smallest positive integer not in S. If, for every i in 1..n-1, a(n-1) + d - a(i) is not in S, then a(n) = a(n-1) + d. Otherwise, let r be the smallest positive integer such that, for every i in 1..n-1, neither a(n-1) + r - a(i) nor a(n-1) + r + d - a(i) is in S; then a(n) = a(n-1) + r and a(n+1) = a(n) + d.

Original entry on oeis.org

0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, 143, 165, 199, 224, 306, 332, 415, 443, 591, 624, 678, 716, 934, 973, 1134, 1174, 1449, 1491, 1674, 1720, 2113, 2161, 2468, 2517, 2855, 2906, 2961, 3245, 3302, 3711, 3772, 4081, 4148, 4603, 4673, 5557, 5628, 5917, 5989

Offset: 0

Thomas Ordowski, Sep 19 2014

Every positive integer is uniquely represented as a difference of two distinct elements of the base set. This is a B_2 sequence.
By the definition of this sequence, with d as the smallest unused difference among terms a(0)..a(n-1), we assign a(n) = a(n-1) + d, provided that this would not cause any difference to be repeated; otherwise, we assign a(n) = a(n-1) + r and a(n+1) = a(n) + d, where r is the smallest integer that allows this assignment of a(n) and a(n+1) without causing any difference to be repeated. Thus, at each step, the smallest unused difference d is either used immediately (as a(n) - a(n-1)) or delayed by one step (and used as a(n+1) - a(n)). In this way, the sequence includes every positive integer as a difference (unlike the Mian-Chowla sequence A005282, which omits differences 33, 88, 98, 99, ...; see A080200).
The set is an optimization of Browkin's base, where r = a(n-1) + 1.
The series Sum_{n>=0} 1/(a(n+1) - a(n)) is divergent.
Conjecture: lim inf_{n->oo} (a(n+1) - a(n))/n = 1/2.

			Given a(0)=0, a(1)=1, a(2)=3, a(3)=7, the differences used are 1,2,3,4,6,7, so d=5, and we can use a(4) = a(3)+d = 7+5 = 12 because appending a(4)=12 to the sequence will result in the differences 12-0=12, 12-1=11, 12-3=9, 12-7=5, none of which had already been used.
Similarly, given a(0)..a(4) = 0,1,3,7,12, the differences used are 1..7,9,11,12, so d=8, and we can use a(5) = a(4)+d = 12+8 = 20 because the resulting differences will be 20, 19, 17, 13, 8, none of which had already been used.
Proceeding as above, we get a(6)=30 and a(7)=44.
Given a(0)..a(7) = 0,1,3,7,12,20,30,44, the differences used are 1..14,17..20,23..24,27,29..30,32,37,41,43..44, so d=15, but we cannot use a(8) = a(7)+d = 44+15 = 59 because the difference 29 would be repeated: 59-30 = 30-1. Thus, we must find the smallest r such that using both a(8) = a(7)+r and a(9) = a(8)+d will not repeat any differences. The smallest such r is 21, so a(8) = a(7)+r = 44+21 = 65 and a(9) = a(8)+d = 65+15 = 80.

Jerzy Browkin, Rozwiązanie pewnego zagadnienia A. Schinzla (Polish) [The solution of a certain problem of A. Schinzel], Roczniki Polskiego Towarzystwa Matematycznego [Annals Polish Mathematical Society], Seria I, Prace Matematyczne III (1959).

Jon E. Schoenfield, Table of n, a(n) for n = 0..6039
Andrew Pollington and Charles Vanden Eynden, The integers as differences of a sequence, Canad. Math. Bull. Vol. 24 (4), 1981 (497-499).
Jon E. Schoenfield, Magma program.

Cf. A001856, where a(1)=1, a(2)=2, a(2n+1)=2*a(2n), a(2n+2) = a(2n+1) + d.
Cf. A005282 (Mian-Chowla sequence), A025582.
Cf. A080200.

a(n) >= A025582(n+1) and for n <= 10 is here equality.

Conjecture: a(n) ~ log(log(n))*A025582(n+1), where A025582(m)+1 = A005282(m) is the Mian-Chowla sequence.

More terms from Jon E. Schoenfield, Jan 18 2015

Edited by Jon E. Schoenfield, Jan 22 2015

A080933 Smallest non-occurring pairwise difference between the elements of a Mian-Chowla sequence (A005282) variant starting with (1,n).

Original entry on oeis.org

33, 49, 26, 15, 30, 19, 44, 13, 38, 50, 54, 58, 44, 46, 25, 20, 45, 10, 13, 84, 38, 15, 71, 33, 35, 54, 31, 16, 57, 10, 42, 26, 15, 14, 33, 14, 15, 32, 34, 16, 25, 28, 25, 16, 36, 16, 16, 25, 28, 40, 16, 31, 33, 28, 15, 31, 15, 22, 31, 33, 15, 21, 49, 51, 28

Offset: 2

Hugo Pfoertner, Feb 24 2003

nonn

For large n the sequence consists of increasingly long runs of "33", which is the first term of A080200 (the smallest non-occurring difference in A005282), with some interspersed other terms. E.g. for n=100000..100100 we have a(n)=88 for n=100005,100021,100041,100087,100099, a(100072)=98 and otherwise a(n)=33.

			a(2)=33 because the smallest non-occurring pairwise difference between the terms of A005282 (starting with 1,2) is A080200(1)=33. a(3)=49 because the smallest non-occurring pairwise difference between the terms of A051788 (starting with 1,3) is A080201(1)=49. a(4)=26 because the smallest non-occurring pairwise difference between the terms of A058335 (starting with 1,4) is A080932(1)=26.

See under A005282.

Hugo Pfoertner, FORTRAN program to create the Mian-Chowla sequence
Hugo Pfoertner, Illustration of the first 1000 terms..

Cf. A005282, A051788, A058335, A080200, A080201, A080932.

cp's OEIS Frontend

A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of 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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A080222 Record-setting differences between adjacent elements of the Mian-Chowla sequence A005282.

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A080932 Non-occurring pairwise differences between the elements of the Mian-Chowla sequence variant A058335.

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A080933 Smallest non-occurring pairwise difference between the elements of a Mian-Chowla sequence (A005282) variant starting with (1,n).

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