A356873 a(n) is the smallest number k such that 2^k+1 has at least n distinct prime factors.

0, 5, 14, 18, 30, 42, 78, 78, 78, 90, 150, 150, 210, 210, 234, 234, 270, 390, 390, 390, 390, 450, 510, 630, 630, 630, 810, 810, 810, 966, 966, 1170, 1170, 1170, 1170, 1170, 1170, 1170

Offset: 1

Author

Alex Ratushnyak, Sep 02 2022

Keywords

• nonn
• hard
• more

Comments

From Jon E. Schoenfield, Sep 04 2022: (Start)

a(39) <= a(40) <= a(41) <= 1530.

a(42) <= a(43) <= a(44) <= 1890.

a(45) <= a(46) <= 2070.

a(47) <= a(48) <= ... <= a(54) = 2730. (End)

Crossrefs

Cf. A046799, A071852.

Cf. A180278, A219108, A356872.

Programs

Mathematica

a[n_] := Block[{k=0}, While[ Length@ FactorInteger[2^k + 1] < n, k++]; k]; Array[a, 12] (* Giovanni Resta, Oct 13 2022 *)

PARI

a(n) = my(k=1); while (omega(2^k+1) < n, k++); k; \\ Michel Marcus, Sep 05 2022

Python

from sympy import factorint, isprime
from itertools import count, islice
def f(n): return 1 if isprime(n) else len(factorint(n))
def agen():
    n = 1
    for k in count(0):
        v = f(2**k+1)
        while v >= n: yield k; n += 1
print(list(islice(agen(), 10))) # Michael S. Branicky, Sep 02 2022

Extensions

a(11)-a(38) from Michael S. Branicky, Sep 02 2022 using A071852