A005087 Number of distinct odd primes dividing n.

0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 0, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Lei Zhou, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A077761, A087436.

Positions of zeros: A000079.

Positions of ones: A336101.

Programs

Haskell

a005087 n = a001221 n + n `mod` 2 - 1 -- Reinhard Zumkeller, Feb 28 2014

Mathematica

nn=100; a=Sum[x^p/(1-x^p), {p, Table[Prime[n],{n,2,nn}]}]; Drop[CoefficientList[Series[a, {x,0,nn}],x],1] (* Geoffrey Critzer, Nov 06 2012 *)
Array[PrimeNu[#] - Boole[EvenQ[#]] &, 102] (* Lei Zhou, Dec 03 2012 *)

PARI

a(n) = if (n%2, omega(n), omega(n)-1); \\ Michel Marcus, Sep 18 2023

Python

from sympy import primefactors
def A005087(n): return len(primefactors(n))+(n&1)-1 # Chai Wah Wu, Jul 07 2022

Sage

def A005087(n) : return len(prime_divisors(n)) + n % 2 - 1
[A005087(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Formula

Additive with a(p^e) = 0 if p = 2, 1 otherwise.

a(n) = A001221(n) - 1 + n mod 2. - Reinhard Zumkeller, Sep 03 2003

O.g.f.: Sum_{p=odd prime} x^p/(1-x^p). - Geoffrey Critzer, Nov 06 2012

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A077761 - 1/2 = -0.238502... . - Amiram Eldar, Sep 28 2023

Extensions

More terms from Reinhard Zumkeller, Sep 03 2003