A074816 a(n) = 3^A001221(n) = 3^omega(n).

1, 3, 3, 3, 3, 9, 3, 3, 3, 9, 3, 9, 3, 9, 9, 3, 3, 9, 3, 9, 9, 9, 3, 9, 3, 9, 3, 9, 3, 27, 3, 3, 9, 9, 9, 9, 3, 9, 9, 9, 3, 27, 3, 9, 9, 9, 3, 9, 3, 9, 9, 9, 3, 9, 9, 9, 9, 9, 3, 27, 3, 9, 9, 3, 9, 27, 3, 9, 9, 27, 3, 9, 3, 9, 9, 9, 9, 27, 3, 9, 3, 9, 3, 27, 9, 9, 9, 9, 3, 27, 9, 9, 9, 9, 9, 9, 3, 9, 9, 9

Offset: 1

Author

Benoit Cloitre, Sep 08 2002

Keywords

• nonn
• mult

Comments

Old name was: a(n) = sum(d|n, tau(d)*mu(d)^2 ).

Terms are powers of 3.

The inverse Mobius transform of A074823, as the Dirichlet g.f. is product_{primes p} (1+2*p^(-s)) and the Dirichlet g.f. of A074816 is product_{primes p} (1+2*p^(-s))/(1-p^(-s)). - R. J. Mathar, Feb 09 2011

If n is squarefree, then a(n) = #{(x, y) : x, y positive integers, lcm (x, y) = n}. See Crandall & Pomerance. - Michel Marcus, Mar 23 2016

References

• Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 2.3 p. 108.

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A034444, A069212, A124508.

Programs

Mathematica

A074816[n_]:=3^PrimeNu[n]; (* Enrique Pérez Herrero, Jun 28 2010 *)

PARI

a(n) = 3^omega(n); \\ Michel Marcus, Mar 23 2016

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Feb 16 2022

Formula

a(n) = 3^m if n is divisible by m distinct primes. i.e., a(n)=3 if n is in A000961; a(n)=9 if n is in A007774 ...

a(n) = 3^A001221(n) = 3^omega(n). Multiplicative with a(p^e)=3. - Vladeta Jovovic, Sep 09 2002.

a(n) = tau_3(rad(n)) = A007425(A007947(n)). - Enrique Pérez Herrero, Jun 24 2010

a(n) = abs(Sum_{d|n} A000005(d^3)*mu(d)). - Enrique Pérez Herrero, Jun 28 2010

a(n) = Sum_{d|n, gcd(d, n/d) = 1} 2^omega(d) (The total number of unitary divisors of the unitary divisors of n). - Amiram Eldar, May 29 2020, Dec 27 2024

a(n) = Sum_{d1|n, d2|n} mu(d1*d2)^2. - Wesley Ivan Hurt, Feb 04 2022

Dirichlet g.f.: zeta(s)^3 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Feb 16 2022

Extensions

Simpler definition at the suggestion of Michel Marcus. - N. J. A. Sloane, Mar 25 2016