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A273711
The Hadamard product of omega(n) and A007875(n).
1
0, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 4, 1, 1, 4, 1, 4, 4, 4, 1, 4, 1, 4, 1, 4, 1, 12, 1, 1, 4, 4, 4, 4, 1, 4, 4, 4, 1, 12, 1, 4, 4, 4, 1, 4, 1, 4, 4, 4, 1, 4, 4, 4, 4, 4, 1, 12, 1, 4, 4, 1, 4, 12, 1, 4, 4, 12, 1, 4, 1, 4, 4, 4, 4, 12, 1, 4, 1, 4, 1, 12, 4
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
Tanay V. Wakhare, On sums involving the number of distinct prime factors functions, arXiv:1604.05671 [math.HO], 2016-2017, Theorem 7.
FORMULA
a(n) = A001221(n)*A007875(n).
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
MAPLE
A273711 := proc(n)
A001221(n)*A007875(n) ;
end proc:
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
Cf. A001221 (omega), A007875, A008683 (mu), A008966.
Sequence in context: A365837 A369669 A155826 * A340227 A351942 A351230
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, May 28 2016
STATUS
approved