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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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]));

Crossrefs

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

Cf. A295662, A295883, A318673, A359472.

Sequence in context: A318381 A341755 A245515 * A327170 A024362 A370256

Adjacent sequences: A366122 A366123 A366124 * A366126 A366127 A366128

Keyword

nonn,easy

Author

Amiram Eldar, Sep 30 2023

Status

approved