A386278 - OEIS

%I #74 Nov 25 2025 17:44:16

%S 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,

%T 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,

%U 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

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

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

%H Andrew Howroyd, <a href="/A386278/b386278.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 *)

%o (PARI) a(n) = factorback(apply(e -> 1 + (e%4) * (3 - (e%4)), factor(n)[, 2]))

%o (Python)

%o from sympy import factorint

%o def A386278(n): return 3**sum(1 for v in factorint(n).values() if 0<v&3<3) # _Chai Wah Wu_, Oct 16 2025

%K nonn,easy,mult

%O 1,2

%A _Werner Schulte_, Oct 12 2025