A386278 Multiplicative sequence a(n) with a(p^e) = 1 + (e mod 4) * (3 - (e mod 4)) for prime p and e >= 0.

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

Offset

1,2

Comments

a(n) = 3^m where m is the number of prime exponents == 1 or 2 mod 4. - Chai Wah Wu, Oct 16 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

Dirichlet g.f.: zeta(4*s) * (zeta(s) / zeta(2*s))^3.

Mathematica

f[p_, e_] := Module[{r = Mod[e, 4]}, 1 + r*(3-r)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 12 2025 *)

Prog

(PARI) a(n) = factorback(apply(e -> 1 + (e%4) * (3 - (e%4)), factor(n)[, 2]))
(Python)
from sympy import factorint
def A386278(n): return 3**sum(1 for v in factorint(n).values() if 0<v&3<3) # Chai Wah Wu, Oct 16 2025

Crossrefs

Sequence in context: A245441 A333793 A007428 * A184099 A074816 A203564

Adjacent sequences: A386275 A386276 A386277 * A386279 A386280 A386281

Keyword

nonn,easy,mult

Author

Werner Schulte, Oct 12 2025

Status

approved