A382969 The excess of the n-th noncubefree number.

2, 3, 2, 2, 4, 2, 3, 2, 2, 5, 3, 3, 3, 2, 4, 2, 3, 3, 2, 2, 6, 2, 2, 4, 2, 4, 3, 2, 3, 2, 2, 5, 3, 3, 4, 4, 2, 3, 4, 2, 2, 7, 2, 2, 3, 2, 5, 2, 2, 3, 2, 5, 4, 2, 3, 2, 2, 2, 4, 3, 3, 2, 2, 2, 6, 3, 4, 3, 2, 4, 2, 5, 2, 5, 2, 2, 3, 2, 4, 4, 2, 3, 3, 3, 8, 2, 2

Offset

1,1

Links

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

Formula

a(n) = A046660(A046099(n)).

a(n) >= 2.

Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = ((Sum_{p prime} 1/(p*(p-1))) - (1/zeta(3)) * (Sum_{p prime} (p-1)/(p^3-1))) / (1-1/zeta(3)) = 3.12223294188308957729... .

Example

a(1) = 2 since the 1st noncubefree number is A046099(1) = 8 = 2^3. It has 3 prime factors when counted with multiplicity, and 1 distinct prime factor, so a(1) = 3 - 1 = 2.

Mathematica

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[Max[e] < 3, Nothing, Total[e] - Length[e]]]; Array[f, 100]

Prog

(PARI) list(lim) = {my(e); for(k = 2, lim, e = factor(k)[, 2]; if(vecmax(e) > 2, print1(vecsum(e) - #e, ", "))); }

Crossrefs

Cf. A002117, A046099, A046660, A136141, A275699, A376366.

Sequence in context: A375341 A368039 A387542 * A393085 A160558 A241019

Adjacent sequences: A382966 A382967 A382968 * A382970 A382971 A382972

Keyword

nonn,easy

Author

Amiram Eldar, Apr 10 2025

Status

approved