%I #13 Feb 03 2024 10:17:04

%S 1,4,4,4,4,16,4,4,4,16,4,16,4,16,16,6,4,16,4,16,16,16,4,16,4,16,4,16,

%T 4,64,4,6,16,16,16,16,4,16,16,16,4,64,4,16,16,16,4,24,4,16,16,16,4,16,

%U 16,16,16,16,4,64,4,16,16,8,16,64,4,16,16,64,4,16,4

%N The number of divisors of the smallest cubefull exponentially odd number that is divisible by n.

%H Amiram Eldar, <a href="/A369757/b369757.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000005(A356192(n)).

%F Multiplicative with a(p^e) = max(e,3) + 1 if e is odd, and e+2 if e is even.

%F a(n) >= A000005(n), with equality if and only if n is cubefull exponentially odd number (A335988).

%F Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 + 3/p^s - 1/p^(2*s) - 3/p^(3*s) + 2/p^(4*s)).

%F From _Vaclav Kotesovec_, Feb 02 2024: (Start)

%F Dirichlet g.f.: zeta(s)^4 * Product_{p prime} (1 + (7*p^(2*s) + 2*p^(3*s) - 6*p^(4*s) - 7*p^s + 2) / ((p^s+1)*p^(5*s))).

%F Sum_{k=1..n} a(k) = c * n*log(n)^3/6 + O(n*log(n)^2), where c = Product_{p prime} (1 - (6*p^4 - 2*p^3 - 7*p^2 + 7*p - 2) / ((p+1)*p^5)) = 0.124604542136592401049820049658828040278... (End)

%t f[p_, e_] := If[OddQ[e], Max[e, 3] + 1, e + 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> if(x%2, max(x, 3) + 1, x + 2), factor(n)[, 2]));

%Y Cf. A000005, A335988, A356192, A365348, A369758, A369759.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Jan 31 2024