A366123 The number of prime factors of the cube root of the largest cube dividing n, counted with multiplicity.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 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, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Sep 30 2023

Keywords

• nonn
• easy

Comments

First differs from A295659 at n = 64.

The number of distinct prime factors of the cube root of the largest cube dividing n is A295659(n).

Links

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

Crossrefs

Cf. A000578, A001222, A002264, A008834, A004709, A053150, A286229, A295659.

Cf. A061704 (number of divisors), A333843 (sum of divisors).

Programs

Mathematica

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

PARI

a(n) = vecsum(apply(x -> x\3, factor(n)[, 2]));

Formula

a(n) = A001222(A053150(n)).

a(n) = A001222(A008834(n))/3.

Additive with a(p^e) = floor(e/3) = A002264(e).

a(n) >= 0, with equality if and only if n is cubefree (A004709).

a(n) <= A001222(n)/3, with equality if and only if n is a positive cube (A000578 \ {0}).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^3-1) = 0.194118... (A286229).