OFFSET
1,6
COMMENTS
Total number of distinct prime factors of the squarefree divisors of n. Inverse Möbius transform of omega(n)*mu(n)^2. - Wesley Ivan Hurt, Jun 17 2023
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Tanay V. Wakhare, On sums involving the number of distinct prime factors functions, arXiv:1604.05671 [math.HO], 2016-2017, Theorem 7.
FORMULA
From Wesley Ivan Hurt, Jun 17 2023: (Start)
a(n) = omega(n)*2^(omega(n)-1).
a(n) = Sum_{d|n} omega(d)*mu(d)^2. (End)
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * P(s, 1), where P(s, c) = Sum_{p prime} 1/(p^s + c) is the shifted prime zeta function (Wakhare, 2016). - Amiram Eldar, Nov 03 2023
MATHEMATICA
Table[EulerPhi[2^PrimeNu[n]]*PrimeNu[n], {n, 1, 50}] (* G. C. Greubel, May 19 2017 *)
PROG
(PARI) a(n)=my(o=omega(n)); o<<(o-1) \\ Charles R Greathouse IV, Jun 07 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, May 28 2016
STATUS
approved