A273711 The Hadamard product of omega(n) and A007875(n).

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

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

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

Adjacent sequences: A273708 A273709 A273710 * A273712 A273713 A273714

Keyword

nonn,easy

Author

R. J. Mathar, May 28 2016

Status

approved