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

1,4

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

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... .

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]

Prog

(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

Crossrefs

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

Cf. A002117, A013661, A253905.

Similar sequences: A368710, A368711, A368713.

Sequence in context: A128257 A198254 A309059 * A293630 A305393 A259154

Adjacent sequences: A368709 A368710 A368711 * A368713 A368714 A368715

Keyword

nonn,easy

Author

Amiram Eldar, Jan 04 2024

Status

approved