A379652 a(1) = 2. 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.

2, 5, 11, 23, 47, 19, 3, 7, 31, 127, 17, 71, 13, 37, 151, 101, 29, 59, 239, 479, 137, 1103, 2207, 883, 2357, 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, 373, 997, 307, 1231

Offset

1,1

Comments

The following are some statistics about how many terms of the sequence are required, so that the first k primes are included:

- The first 10^2 terms include the first 17 primes.

- The first 10^3 terms include the first 131 primes.

- The first 10^4 terms include the first 808 primes.

- The first 10^5 terms include the first 4397 primes.

- The first 10^6 terms include the first 35801 primes.

- The first 10^7 terms include the first 253682 primes.

Conjecture: this sequence is a permutation of the primes.

If we start with 1 instead of 2 we get A379727. - N. J. A. Sloane, Dec 31 2024

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

a(6) is 19 because the prime factors of c=2*a(5)+1 (i.e., 95) are 5 and 19, and 5 already appears in the sequence as a(2).
a(9) is 31 because the prime factors of c=2*a(8)+1 (i.e., 15) are 3 and 5 which already appear in the sequence as a(7) and a(2). The next value of c (i.e., 2*c+1) is 31, which is prime and does not already appear in the sequence.

Mathematica

c[_] := True; j = 2; c[2] = 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 29 2024 *)

Prog

(Python)
from sympy import primefactors
seq = [2]
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)

Crossrefs

Cf. A031439, A072268, A131200, A174162, A379648.

Sequence in context: A349411 A347309 A174162 * A340799 A186253 A226462

Adjacent sequences: A379649 A379650 A379651 * A379653 A379654 A379655

Keyword

nonn

Author

Robert C. Lyons, Dec 28 2024

Status

approved