A082476 a(n) = Sum_{d|n} mu(d)^2*tau(d)^2.

1, 5, 5, 5, 5, 25, 5, 5, 5, 25, 5, 25, 5, 25, 25, 5, 5, 25, 5, 25, 25, 25, 5, 25, 5, 25, 5, 25, 5, 125, 5, 5, 25, 25, 25, 25, 5, 25, 25, 25, 5, 125, 5, 25, 25, 25, 5, 25, 5, 25, 25, 25, 5, 25, 25, 25, 25, 25, 5, 125, 5, 25, 25, 5, 25, 125, 5, 25, 25, 125, 5, 25, 5, 25, 25, 25, 25, 125

Offset: 1

Author

Benoit Cloitre, Apr 27 2003

Keywords

• mult
• nonn

Comments

More generally : sum(d|n, mu(d)^2*tau(d)^m) = (2^m+1)^omega(n).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000005, A000351, A001221, A007425, A007947, A008683, A061200, A074816.

Programs

Mathematica

tau[1, n_] := 1; SetAttributes[tau, Listable];
tau[k_, n_] := Plus @@ (tau[k - 1, Divisors[n]]) /; k > 1;
A082476[n_] := Abs[DivisorSum[n, MoebiusMu[ # ]*tau[3, #^2] &]]; (* Enrique Pérez Herrero, Mar 29 2010 *)
(* or more easy *)
A082476[n_] := 5^PrimeNu[n] (* Enrique Pérez Herrero, Mar 29 2010 *)

PARI

a(n)=5^omega(n)

PARI

for(n=1, 100, print1(direuler(p=2, n, (4*X+1)/(1-X))[n], ", ")) \\ Vaclav Kotesovec, Feb 28 2023

Formula

a(n) = 5^omega(n); multiplicative with a(p^e)=5.

a(n) = abs(sum(d|n, mu(d)*tau_3(d^2))), where tau_3 is A007425. - Enrique Pérez Herrero, Mar 29 2010

a(n) = tau_5(rad(n)) = A061200(A007947(n)). - Enrique Pérez Herrero, Jun 24 2010

a(n) = A000351(A001221(n)). - Antti Karttunen, Jul 26 2017

From Vaclav Kotesovec, Feb 28 2023: (Start)

Dirichlet g.f.: Product_{primes p} (1 + 5/(p^s - 1)).

Dirichlet g.f.: zeta(s)^5 * Product_{primes p} (1 - 10/p^(2*s) + 20/p^(3*s) - 15/p^(4*s) + 4/p^(5*s)), (with a product that converges for s=1). (End)