A133097 -id:A133097 - cp's OEIS frontend

A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

1, 2, 3, 5, 8, 13, 21, 30, 39, 53, 74, 95, 128, 152, 182, 212, 258, 316, 374, 413, 476, 531, 546, 608, 717, 798, 862, 965, 1060, 1161, 1307, 1386, 1435, 1556, 1722, 1834, 1934, 2058, 2261, 2497, 2699, 2874, 3061, 3197, 3332, 3629, 3712, 3868, 4140, 4447, 4640

Offset: 1

Dan Hoey

nonn

a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j)-a(k) for distinct i and j with 1 <= i,j,k <= n-1. [Comment corrected by Jean-Paul Delahaye, Oct 02 2020.]

Klaus Brockhaus and Dan Hoey, Table of n, a(n) for n = 1..2839
Index entries for B_2 sequences.

Cf. A010672.
Cf. A033627, A003278, A026471, A066720, A075122, A075123, A133097.
Cf. A036241, A349777.

from itertools import islice
def agen(): # generator of terms
    aset, sset, k = set(), set(), 0
    while True:
        k += 1
        while any(k+an in sset for an in aset): k += 1
        yield k; sset.update(k+an for an in aset); aset.add(k)
print(list(islice(agen(), 51))) # Michael S. Branicky, Feb 05 2023

a(n) = A010672(n-1)+1.

A133096 a(n) = A079848(n) - A062294(n).

Original entry on oeis.org

0, 0, 0, 4, 12, 20, 18, 50, 34, 66, 80, 70, 120, 76, 90, 198, 140, 70, 110, 156, 100, 164, 10, -6, 198, 66, 258, 280, 732, 390, 594, 342, 270, 314, 210, 402, 252, 246, -28, -204, -224, 234, 14, 528, 500, 850, 1110, 942, 1014, 1542, 1906, 1416, 1034, 1454, 970, 982, 1518

Offset: 1

Klaus Brockhaus, Sep 17 2007

sign

A079848 is the sequence of smallest primes such that the pairwise sums of not necessarily distinct elements are all distinct, whereas A062294 is the sequence of smallest primes such that the pairwise sums of distinct elements are all distinct.

			a(6) = A079848(6) - A062294(6) = 37 - 17 = 20.

Klaus Brockhaus, Table of n, a(n) for n = 1..3600

Cf. A062294, A079848, A133097.

from collections import deque
from itertools import islice
from sympy import nextprime
def A133096_gen(): # generator of terms
    aset2, alist, bset2, blist, aqueue, bqueue, k = set(), [], set(), [], deque(), deque(), 1
    while (k:=nextprime(k)):
        cset2 = set()
        for a in alist:
            if (m:=k-a) 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()
A133096_list = list(islice(A133096_gen(),30)) # Chai Wah Wu, Sep 11 2023

cp's OEIS Frontend

A011185 A B_2 sequence: a(n) = least value such that sequence increases and pairwise sums of distinct elements are all distinct.

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