A061784 -id:A061784 - cp's OEIS frontend

A061781 Number of distinct sums p(i) + p(j) for 1<=i<=j<=n, p(k) = k-th prime.

1, 3, 6, 9, 13, 17, 21, 25, 29, 33, 39, 44, 50, 54, 59, 63, 67, 75, 80, 86, 91, 95, 101, 107, 114, 120, 126, 131, 136, 140, 148, 154, 160, 168, 174, 180, 187, 192, 199, 205, 211, 219, 224, 231, 237, 242, 249, 255, 264, 270, 278, 283, 289, 296, 302, 306, 310, 319

Offset: 1

Labos Elemer, Jun 22 2001

nonn

			Let P(n) = {2, 3, .., p_n} be the set of the first n primes. Construct S(n) = {p+q : p,q in P}. If every sum p+q were distinct, then |S(n)| would be n*(n+1)/2 = A000217(n). But in reality, for n >= 4, certain sums occur more than once. a(n) = |S(n)|. For example, P(6) = {2, 3, 5, 7, 11, 13} yields S(6) = {4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26}. Thus, a(6) = 17.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A000217, A061784.

f[x_] := Prime[x]
Table[Length[Union[Flatten[Table[f[u]+f[w], {w, 1, m}, {u, 1, m}]]]], {m, 1, 75}]

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

A061781 Number of distinct sums p(i) + p(j) for 1<=i<=j<=n, p(k) = k-th prime.

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