A074816 - OEIS

%I #47 Dec 27 2024 04:11:35

%S 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,

%T 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,

%U 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

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

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

%C Terms are powers of 3.

%C 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

%C 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

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

%H R. Zumkeller, <a href="/A074816/b074816.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 ...

%F a(n) = 3^A001221(n) = 3^omega(n). Multiplicative with a(p^e)=3. - _Vladeta Jovovic_, Sep 09 2002.

%F a(n) = tau_3(rad(n)) = A007425(A007947(n)). - _Enrique Pérez Herrero_, Jun 24 2010

%F a(n) = abs(Sum_{d|n} A000005(d^3)*mu(d)). - _Enrique Pérez Herrero_, Jun 28 2010

%F 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

%F a(n) = Sum_{d1|n, d2|n} mu(d1*d2)^2. - _Wesley Ivan Hurt_, Feb 04 2022

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

%t A074816[n_]:=3^PrimeNu[n]; (* _Enrique Pérez Herrero_, Jun 28 2010 *)

%o (PARI) a(n) = 3^omega(n); \\ _Michel Marcus_, Mar 23 2016

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 16 2022

%Y Cf. A001221, A034444, A069212, A124508.

%K nonn,mult

%O 1,2

%A _Benoit Cloitre_, Sep 08 2002

%E Simpler definition at the suggestion of _Michel Marcus_. - _N. J. A. Sloane_, Mar 25 2016