A389713 a(1) = 3, a(2) = 5, a(n) is the smallest prime such that a(n) - a(n-1) + 1 is in the sequence.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset

1,1

Comments

It is unknown whether this sequence is infinite.

References

Paul Erdős and Ronald Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Thomas Bloom, Problem #472, Erdős Problems.

Example

a(3) = a(1) + a(2) - 1 = 7.
The first prime missing from this sequence is 97, because 97 - 89 + 1 = 9, which is not prime.

Mathematica

a={3, 5}; For[i=3, i<100, i++, AppendTo[a, Module[{p=NextPrime@a[[-1]]}, While[Not@MemberQ[a, p-a[[-1]]+1], p=NextPrime[p]]; p]]]; a

Prog

(Python)
from gmpy2 import next_prime
from itertools import islice
def agen():
    aset, an = {3}, 5
    yield 3
    while True:
        yield an
        aset.add(an)
        p, ub = next_prime(an), 2*an - 1
        while p - an + 1 not in aset and p <= ub:
            p = next_prime(p)
        if p > ub:
            print("Sequence is finite")
            return
        an = int(p)
print(list(islice(agen(), 58))) # Michael S. Branicky, Jan 09 2026

Crossrefs

Sequence in context: A240699 A065380 A211075 * A038134 A215697 A322184

Adjacent sequences: A389710 A389711 A389712 * A389714 A389715 A389716

Keyword

nonn

Author

Elijah Beregovsky, Jan 08 2026

Status

approved