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

1,15

Links

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

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

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 *)

Prog

(Sage)
def A005087(n) : return len(prime_divisors(n)) + n % 2 - 1
[A005087(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012
(Haskell)
a005087 n = a001221 n + n `mod` 2 - 1 -- Reinhard Zumkeller, Feb 28 2014
(Python)
from sympy import primefactors
def A005087(n): return len(primefactors(n))+(n&1)-1 # Chai Wah Wu, Jul 07 2022
(PARI) a(n) = if (n%2, omega(n), omega(n)-1); \\ Michel Marcus, Sep 18 2023

Crossrefs

Cf. A001221, A077761, A087436.

Positions of zeros: A000079.

Positions of ones: A336101.

Sequence in context: A284203 A341594 A368774 * A050332 A369258 A337930

Adjacent sequences: A005084 A005085 A005086 * A005088 A005089 A005090

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, Sep 03 2003

Status

approved