A379727 a(1) = 1. For n > 1, a(n) = smallest prime factor of c=2*a(n-1)+1 that is not in {a(1), ..., a(n-1)}; if all prime factors of c are in {a(1), ..., a(n-1)}, then we try the next value of c, which is 2*c+1; and so on.

1, 3, 7, 5, 11, 23, 47, 19, 13, 37, 151, 101, 29, 59, 17, 71, 41, 83, 167, 67, 271, 181, 727, 97, 1567, 6271, 113, 227, 911, 1823, 521, 149, 599, 109, 73, 197, 79, 53, 107, 43, 563, 347, 139, 31, 127, 89, 179, 359, 719, 1439, 2879, 443, 887, 7103, 14207, 5683, 421, 281, 1289, 2579, 607, 1621, 499, 1999, 5333, 10667, 251, 503, 733, 163, 131, 263, 211, 3391, 13567, 7753, 1723, 383, 307, 1231, 821, 173, 293

Offset: 1

Author

N. J. A. Sloane, Dec 31 2024

Keywords

• nonn

Comments

If we start with a(1) = 2, we get A379652.

Links

• Robert C. Lyons, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A379652, A379649.

Programs

Mathematica

c[_] := True; j = 1; c[1] = False;
{j}~Join~Reap[Do[
  m = 2*j + 1;
  While[
    Set[k, SelectFirst[FactorInteger[m][[All, 1]], c]]; !
      IntegerQ[k], m = 2*m + 1]; c[k] = False;
j = Sow[k], {120}] ][[-1, 1]] (* Michael De Vlieger, Dec 31 2024 *)

Python

from sympy import primefactors
seq = [1]
seq_set = set(seq)
max_seq_len=100
while len(seq) <= max_seq_len:
    c = seq[-1]
    done = False
    while not done:
        c = 2*c+1
        factors = primefactors(c)
        for factor in factors:
            if factor not in seq_set:
                seq.append(factor)
                seq_set.add(factor)
                done = True
                break
print(seq) # Robert C. Lyons, Jan 01 2025