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

1,3

Comments

Bisection of A001221.

Links

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

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)

Example

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.

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 *)

Prog

(PARI) for(n=1, 50, print1(omega(2*n), ", ")) \\ G. C. Greubel, May 21 2017
(Magma) [#PrimeDivisors(2*n): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Crossrefs

Cf. A001221, A077761, A092523.

Sequence in context: A201219 A254315 A080942 * A246600 A068068 A193523

Adjacent sequences: A099809 A099810 A099811 * A099813 A099814 A099815

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 19 2004

Extensions

More terms from Emeric Deutsch, Mar 10 2005

Status

approved