A046798 Number of divisors of 2^n + 1.

2, 2, 2, 3, 2, 4, 4, 4, 2, 8, 6, 4, 4, 4, 8, 12, 2, 4, 16, 4, 4, 12, 8, 4, 8, 16, 16, 20, 4, 8, 48, 4, 4, 24, 16, 32, 16, 8, 16, 12, 4, 8, 64, 4, 8, 64, 32, 8, 8, 8, 64, 48, 8, 8, 64, 48, 8, 24, 8, 16, 16, 4, 32, 64, 4, 64, 64, 8, 12, 24, 96, 8, 32, 8, 32, 96, 16, 64, 768, 4, 8, 192, 32, 64

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Comments

a(n) is odd iff n = 3, as a consequence of the Catalan-Mihăilescu theorem. - Bernard Schott, Oct 05 2021

Examples

			a(7)=4, because 2^7 + 1 = 129 has 4 divisors.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..1128 (terms 0..500 from T. D. Noe, terms 501..1062 from Amiram Eldar, term 1108 from Tyler Busby)
• Wikipedia, Catalan's conjecture.

Crossrefs

Cf. A000005, A000051, A002587, A002589, A046801, A054992, A057957, A059886, A274906, A344897.

Programs

Magma

[NumberOfDivisors(2^n+1): n in [0..100]]; // Vincenzo Librandi, Feb 05 2018

Maple

a:= n-> numtheory[tau](2^n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 23 2021

Mathematica

A046798[n_IntegerQ]:=DivisorSigma[0,1+2^n]; (* Enrique Pérez Herrero, Nov 09 2010 *)
DivisorSigma[0, 1 + 2^#] & /@ Range[0, 83] (* Jayanta Basu, Jun 29 2013 *)
Table[DivisorSigma[0, 2^n + 1], {n, 0, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

PARI

a(n) = numdiv(2^n+1); \\ Michel Marcus, Mar 18 2017

Python

from sympy.ntheory import divisor_count
def A046798(n): return divisor_count(2**n + 1) # Indranil Ghosh, Mar 18 2017

Formula

a(n) = A000005(A000051(n)). - Michel Marcus, Mar 18 2017