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Number of prime factors of the squarefree numbers: omega(A005117(n)).
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%I #34 Sep 21 2024 14:36:33

%S 0,1,1,1,2,1,2,1,1,2,2,1,1,2,2,1,2,1,3,1,2,2,2,1,2,2,1,3,1,2,1,2,1,2,

%T 2,2,1,1,2,2,3,1,2,3,1,1,2,2,3,1,2,1,2,2,2,1,2,2,2,2,1,1,3,1,3,2,1,1,

%U 3,2,1,3,2,2,2,2,2,1,2,3,1,2,2,1,3,1,2

%N Number of prime factors of the squarefree numbers: omega(A005117(n)).

%C For n > 1: length of row n in A265668. - _Reinhard Zumkeller_, Dec 13 2015

%H Reinhard Zumkeller, <a href="/A072047/b072047.txt">Table of n, a(n) for n = 1..10000</a>

%H Rafael Jakimczuk and Matilde Lalín, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Lalin/lalin2.html">The Number of Prime Factors on Average in Certain Integer Sequences</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

%F a(n) = A001221(A005117(n)) = A001222(A005117(n)).

%F Sum_{A005117(k) <= x} a(k) = (1/zeta(2))*x*log(log(x)) + O(x) (Jakimczuk and Lalín, 2022). - _Amiram Eldar_, Feb 18 2023, corrected Sep 21 2024

%t PrimeOmega[Select[Range[200],SquareFreeQ]] (* _Harvey P. Dale_, May 14 2011 *)

%o (Haskell)

%o a072047 n = a072047_list !! (n-1)

%o a072047_list = map a001221 $ a005117_list

%o -- _Reinhard Zumkeller_, Aug 08 2011

%o (PARI) apply(omega, select(issquarefree, [1..200])) \\ _Michel Marcus_, Nov 25 2022

%o (Python)

%o from math import isqrt

%o from sympy import mobius, primenu

%o def A072047(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))

%o def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o return primenu(bisection(f)) # _Chai Wah Wu_, Aug 31 2024

%Y Cf. A072048.

%Y Cf. A001221, A001222, A013661, A005117, A265668.

%K nonn,nice

%O 1,5

%A _Reinhard Zumkeller_, Jun 09 2002