A046799 -id:A046799 - cp's OEIS frontend

A071852 Smallest k such that 2^k + 1 has exactly n distinct prime factors.

1, 5, 14, 18, 30, 42, 99, 114, 78, 90, 175, 150, 324, 210, 315, 234, 270, 585, 405, 765, 390, 450, 510, 1150, 690, 630, 930, 858, 810, 1155, 966, 1386

Offset: 1

Benoit Cloitre, Jun 09 2002

a(33) > 1500; a(34) = 1365; a(35) = 1350; a(38) = 1170; a(41) = 1530. - Max Alekseyev, Oct 14 2012

a(33) <= 1782; a(36) <= 1710; a(42) <= 2142; a(43) <= 2394; a(44) <= 1890; a(45) <= 2310; a(46) <= 2070. - Jon E. Schoenfield, Sep 03 2022

Cf. A046799.

For[n = 1, n < 15, n++, k := 1; While[Not[Length[FactorInteger[2^k + 1]] == n], k++ ]; Print[k]] (* Stefan Steinerberger, Apr 09 2006 *)

for(n=1,10,s=1; while(abs(omega(2^s+1)-n)>0,s++); print1(s,","))

a(n) = min (k : A046799(k) = n ).

175 and 150 from Erich Friedman, Aug 08 2005
a(13)-a(23) from Donovan Johnson, Apr 22 2008
a(24)-a(32) from Max Alekseyev, Oct 14 2012

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

Original entry on oeis.org

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

Alex Ratushnyak, Sep 02 2022

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)

Cf. A046799, A071852.

Cf. A180278, A219108, A356872.

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

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

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

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

cp's OEIS Frontend

A071852 Smallest k such that 2^k + 1 has exactly n distinct prime factors.

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