A368712 The maximal exponent in the prime factorization of the cubefree numbers.

0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 04 2024

Keywords

• nonn
• easy

Comments

The asymptotic density of occurrences of 1 is zeta(3)/zeta(2) = 0.730762... (A253905), and the asymptotic density of occurrences of 2 is 1 - zeta(3)/zeta(2) = 0.269237... .

Links

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

Crossrefs

Cf. A004709, A005117, A008966, A013929, A033150, A051903, A067259.

Cf. A002117, A013661, A253905.

Similar sequences: A368710, A368711, A368713.

Programs

Mathematica

s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]
(* or *)
f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e < 3, e, Nothing]]; f[1] = 0; Array[f, 100]

PARI

lista(kmax) = {my(e); print1(0, ", "); for(k = 2, kmax, e = vecmax(factor(k)[,2]); if(e < 3, print1(e, ", ")));}

Python

from sympy import mobius, integer_nthroot, factorint
def A368712(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return max(factorint(m).values(),default=0) # Chai Wah Wu, Aug 12 2024

Formula

a(n) = A051903(A004709(n)).

a(n) = 2 - A008966(A004709(n)) for n >= 2.

Except for n = 1, a(n) = 1 or 2.

a(n) = 1 if and only if A004709(n) is squarefree (A005117).

a(n) = 2 if and only if A004709(n) > 1 and is nonsquarefree (A013929), i.e., A004709(n) is in A067259.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2 - zeta(3)/zeta(2) = 2 - A253905 = 1.269237030598... .