A099812 Number of distinct primes dividing 2n (i.e., omega(2n)).

1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 3, 1, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 4

Offset: 1

Author

N. J. A. Sloane, Nov 19 2004

Keywords

• nonn
• easy

Comments

Bisection of A001221.

Examples

			a(6) = 2 because 12 = 2*2*3 has 2 distinct prime divisors.
a(15) = 3 because 30 = 2*3*5 has 3 distinct prime divisors.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001221, A077761, A092523.

Programs

Magma

[#PrimeDivisors(2*n): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Maple

with(numtheory): omega:=proc(n) local div,A,j: div:=divisors(n): A:={}: for j from 1 to tau(n) do if isprime(div[j])=true then A:=A union {div[j]} else A:=A fi od: nops(A) end: seq(omega(2*n),n=1..130); # Emeric Deutsch, Mar 10 2005

Mathematica

Table[PrimeNu[2*n], {n,1,50}] (* G. C. Greubel, May 21 2017 *)

PARI

for(n=1,50, print1(omega(2*n), ", ")) \\ G. C. Greubel, May 21 2017

Formula

From Amiram Eldar, Sep 21 2024: (Start)

a(n) = A001221(2*n).

a(n) = omega(n) + 1 if n is odd, and a(n) = omega(n) if n is even.

Sum_{k=1..n} a(k) = n * (log(log(n)) + B + 1/2) + O(n/log(n)), where B is Mertens's constant (A077761). (End)

Extensions

More terms from Emeric Deutsch, Mar 10 2005