A133096 -id:A133096 - cp's OEIS frontend

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

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

cp's OEIS Frontend

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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