A025582 A B_2 sequence: a(n) is the least value such that sequence increases and pairwise sums of elements are all distinct.

0, 1, 3, 7, 12, 20, 30, 44, 65, 80, 96, 122, 147, 181, 203, 251, 289, 360, 400, 474, 564, 592, 661, 774, 821, 915, 969, 1015, 1158, 1311, 1394, 1522, 1571, 1820, 1895, 2028, 2253, 2378, 2509, 2779, 2924, 3154, 3353, 3590, 3796, 3997, 4296, 4432, 4778, 4850

Offset

1,3

Comments

a(n) is also the least value such that sequence increases and pairwise differences of distinct elements are all distinct.

Links

Adam Reichert, Table of n, a(n) for n = 1..10000 (terms 1..1000 from David W. Wilson)

Alon Amit, What are some interesting proofs using transfinite induction?, Quora, Nov 15 2014.

Michael K.-H. Kiessling and David J. Wales, A note on the minimal pairwise distance in optimal Lennard-Jones N-body clusters, Molec. Phys. (2025). See p. 10. See also arXiv:2511.15008 [physics.atm-clus], 2025. See p. 11.

Adam Reichert, Python

Eric Weisstein's World of Mathematics, B2 Sequence.

LiJun Zhang, Bing Li, and LeeTang Cheng, Constructions of QC LDPC codes based on integer sequences, Science China Information Sciences, June 2014, Volume 57, Issue 6, pp 1-14.

Index entries for B_2 sequences.

Formula

a(n) = A005282(n) - 1. - Tyler Busby, Mar 16 2024

Example

After 0, 1, a(3) cannot be 2 because 2+0 = 1+1, so a(3) = 3.

Prog

(SageMath)
def A025582_list(n):
    a = [0]
    psums = set([0])
    while len(a) < n:
        a += [next(k for k in IntegerRange(a[-1]+1, infinity) if not any(i+k in psums for i in a+[k]))]
        psums.update(set(i+a[-1] for i in a))
    return a[:n]
print(A025582_list(20))
# D. S. McNeil, Feb 20 2011
(Python)
from itertools import count, islice
def A025582_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(0):
        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
A025582_list = list(islice(A025582_gen(), 20)) # Chai Wah Wu, Sep 01 2023

Crossrefs

Row 2 of A365515.

See A011185 for more information.

A010672 is a similar sequence, but there the pairwise sums of distinct elements are all distinct.

Sequence in context: A173256 A330285 A002049 * A247556 A337656 A029452

Adjacent sequences: A025579 A025580 A025581 * A025583 A025584 A025585

Keyword

nonn,changed

Author

Dan Hoey

Status

approved