login
A366125
The number of prime factors of the cube root of the largest unitary divisor of n that is a cube (A366126), counted with multiplicity.
3
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, 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, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
OFFSET
1,64
COMMENTS
First differs from A295662 at n = 32, and from A295883, A318673, A359472 and A366124 at n = 64.
One third of the sum of exponents that are divisible by 3 in the prime factorization of n.
LINKS
FORMULA
a(n) = A001222(A366126(n))/3.
Additive with a(p^e) = A175676(e) = e if e is divisible by 3, and 0 otherwise.
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... .
MATHEMATICA
f[p_, e_] := If[Divisible[e, 3], e/3, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecsum(apply(x -> if(!(x%3), x/3, 0), factor(n)[, 2]));
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Sep 30 2023
STATUS
approved