

A046800


Number of distinct prime factors of 2^n1.


14



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
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OFFSET

0,5


LINKS

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


FORMULA

a(n) = sum{dn} 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^n1.  Charles R Greathouse IV, Apr 12 2012
a(n) = A001221(2^n1).  R. J. Mathar, Nov 10 2017


EXAMPLE

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


MAPLE

A046800 := proc(n)
if n <= 1 then
0;
else
numtheory[factorset](2^n1) ;
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 *)


PROG

(PARI) a(n)=omega(2^n1) \\ Charles R Greathouse IV, Nov 17 2014


CROSSREFS

Length of row n of A060443.
Cf. A000225, A046051 (number of prime factors, with repetition, of 2^n1), A086251.
Sequence in context: A191372 A185316 A053279 * A027350 A029327 A079135
Adjacent sequences: A046797 A046798 A046799 * A046801 A046802 A046803


KEYWORD

nonn


AUTHOR

Labos Elemer


EXTENSIONS

Edited by T. D. Noe, Jul 14 2003


STATUS

approved



