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%I #45 Sep 28 2023 04:22:32
%S 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,
%T 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,
%U 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
%N Number of distinct odd primes dividing n.
%H Lei Zhou, <a href="/A005087/b005087.txt">Table of n, a(n) for n = 1..10000</a>
%F Additive with a(p^e) = 0 if p = 2, 1 otherwise.
%F a(n) = A001221(n) - 1 + n mod 2. - _Reinhard Zumkeller_, Sep 03 2003
%F O.g.f.: Sum_{p=odd prime} x^p/(1-x^p). - _Geoffrey Critzer_, Nov 06 2012
%F 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
%t 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 *)
%t Array[PrimeNu[#] - Boole[EvenQ[#]] &, 102] (* _Lei Zhou_, Dec 03 2012 *)
%o (Sage)
%o def A005087(n) : return len(prime_divisors(n)) + n % 2 - 1
%o [A005087(n) for n in (1..80)] # Peter Luschny, Feb 01 2012
%o (Haskell)
%o a005087 n = a001221 n + n `mod` 2 - 1 -- _Reinhard Zumkeller_, Feb 28 2014
%o (Python)
%o from sympy import primefactors
%o def A005087(n): return len(primefactors(n))+(n&1)-1 # _Chai Wah Wu_, Jul 07 2022
%o (PARI) a(n) = if (n%2, omega(n), omega(n)-1); \\ _Michel Marcus_, Sep 18 2023
%Y Cf. A001221, A077761, A087436.
%Y Positions of zeros: A000079.
%Y Positions of ones: A336101.
%K nonn,easy
%O 1,15
%A _N. J. A. Sloane_
%E More terms from _Reinhard Zumkeller_, Sep 03 2003
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