A344337 a(n) = 9^omega(n), where omega(n) is the number of distinct primes dividing n.

1, 9, 9, 9, 9, 81, 9, 9, 9, 81, 9, 81, 9, 81, 81, 9, 9, 81, 9, 81, 81, 81, 9, 81, 9, 81, 9, 81, 9, 729, 9, 9, 81, 81, 81, 81, 9, 81, 81, 81, 9, 729, 9, 81, 81, 81, 9, 81, 9, 81, 81, 81, 9, 81, 81, 81, 81, 81, 9, 729, 9, 81, 81, 9, 81, 729, 9, 81, 81, 729, 9, 81, 9, 81, 81, 81

Offset: 1

Author

Seiichi Manyama, May 15 2021

Keywords

• nonn
• mult

Crossrefs

k^omega(n): A034444 (k=2), A074816 (k=3), A082476 (k=5), this sequence (k=9).

Cf. A000005, A001019, A001221.

Programs

Mathematica

Table[9^PrimeNu[n], {n, 1, 100}] (* Amiram Eldar, May 15 2021 *)

PARI

a(n) = 9^omega(n);

PARI

a(n) = prod(k=1, #f=factor(n)[, 2], 9);

PARI

a(n) = sumdiv(n, d, moebius(d)^2*numdiv(d)^3);

Formula

a(n) = A001019(A001221(n)).

Multiplicative with a(p^e) = 9.

a(n) = Sum_{d|n} mu(d)^2 * tau(d)^3.

Dirichlet g.f.: Product_{p prime} (1 + 9/(p^s-1)). - Amiram Eldar, Sep 19 2023