A046800 Number of distinct prime factors of 2^n-1.

0, 0, 1, 1, 2, 1, 2, 1, 3, 2, 3, 2, 4, 1, 3, 3, 4, 1, 4, 1, 5, 3, 4, 2, 6, 3, 3, 3, 6, 3, 6, 1, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 6, 3, 7, 6, 4, 3, 9, 2, 7, 5, 7, 3, 6, 6, 8, 4, 6, 2, 11, 1, 3, 6, 7, 3, 8, 2, 7, 4, 9, 3, 12, 3, 5, 7, 7, 4, 7, 3, 9, 6, 5, 2, 12, 3, 5, 6, 10, 1, 11, 5, 9, 3, 6, 5, 12, 2, 5, 8, 12, 2

Offset: 0

Author

Labos Elemer

Keywords

• nonn

Examples

			a(6) = 2 because 63 = 3*3*7 has 2 distinct prime factors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1206 (terms 1..500 from T. D. Noe)
• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• Abílio Lemos and Ady Cambraia Junior, On the number of prime factors of Mersenne numbers (2016)

Crossrefs

Length of row n of A060443.

Cf. A000225, A046051 (number of prime factors, with repetition, of 2^n-1), A086251.

Programs

Maple

A046800 := proc(n)
    if n <= 1 then
        0;
    else
        numtheory[factorset](2^n-1) ;
        nops(%) ;
    end if;
end proc:
seq(A046800(n),n=0..100) ; # R. J. Mathar, Nov 10 2017

Mathematica

Table[Length[ FactorInteger [ 2^n -1 ] ], {n, 0, 100}]
Join[{0},PrimeNu/@(2^Range[110]-1)] (* Harvey P. Dale, Mar 09 2015 *)

PARI

a(n)=omega(2^n-1) \\ Charles R Greathouse IV, Nov 17 2014

Formula

a(n) = Sum_{d|n} A086251(d), Mobius transform of A086251.

a(n) < 0.7 * n; the constant 0.7 cannot be improved below log 2 using only the size of 2^n-1. - Charles R Greathouse IV, Apr 12 2012

a(n) = A001221(2^n-1). - R. J. Mathar, Nov 10 2017

Extensions

Edited by T. D. Noe, Jul 14 2003