

A074816


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


7



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
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OFFSET

1,2


COMMENTS

Old name was: a(n) = sum(dn, 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))/(1p^(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

R. Crandall and C. 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


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(dn, A000005(d^3)*mu(d))).  Enrique Pérez Herrero, Jun 28 2010
a(n) = Sum_{dn, gcd(d, n/d) = 1} 2^omega(d).  Amiram Eldar, May 29 2020


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A001221, A034444, A124508.
Sequence in context: A333793 A007428 A184099 * A203564 A111575 A161836
Adjacent sequences: A074813 A074814 A074815 * A074817 A074818 A074819


KEYWORD

nonn,mult


AUTHOR

Benoit Cloitre, Sep 08 2002


EXTENSIONS

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


STATUS

approved



