A366125 - OEIS

%I #9 Sep 30 2023 21:57:26

%S 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,

%T 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,2,0,0,0,0,

%U 0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1

%N The number of prime factors of the cube root of the largest unitary divisor of n that is a cube (A366126), counted with multiplicity.

%C First differs from A295662 at n = 32, and from A295883, A318673, A359472 and A366124 at n = 64.

%C One third of the sum of exponents that are divisible by 3 in the prime factorization of n.

%H Amiram Eldar, <a href="/A366125/b366125.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A001222(A366126(n))/3.

%F Additive with a(p^e) = A175676(e) = e if e is divisible by 3, and 0 otherwise.

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} p^2*(p-1)/(p^3-1)^2 = 0.35687351842962928035... .

%t f[p_, e_] := If[Divisible[e, 3], e/3, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecsum(apply(x -> if(!(x%3), x/3, 0), factor(n)[, 2]));

%Y Cf. A001222, A175676, A350386, A366124, A366126.

%Y Cf. A295662, A295883, A318673, A359472.

%K nonn,easy

%O 1,64

%A _Amiram Eldar_, Sep 30 2023