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

1,64

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

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).

Mathematica

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

Prog

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

Crossrefs

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

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

Sequence in context: A244413 A318655 A056626 * A290081 A347706 A348381

Adjacent sequences: A366120 A366121 A366122 * A366124 A366125 A366126

Keyword

nonn,easy

Author

Amiram Eldar, Sep 30 2023

Status

approved