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The maximal exponent in the prime factorization of the cubefree numbers.
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%I #16 Aug 12 2024 16:35:38

%S 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,

%T 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,

%U 1,1,2,1,1,1,1,2,1,2,1,1,1,1,2,2,2,1,1

%N The maximal exponent in the prime factorization of the cubefree numbers.

%C 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... .

%H Amiram Eldar, <a href="/A368712/b368712.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%t s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[100], Max[FactorInteger[#][[;; , 2]]] < 3 &]

%t (* or *)

%t f[n_] := Module[{e = Max @@ FactorInteger[n][[;; , 2]]}, If[e < 3, e, Nothing]]; f[1] = 0; Array[f, 100]

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

%o (Python)

%o from sympy import mobius, integer_nthroot, factorint

%o def A368712(n):

%o def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o m, k = n, f(n)

%o while m != k:

%o m, k = k, f(k)

%o return max(factorint(m).values(),default=0) # _Chai Wah Wu_, Aug 12 2024

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

%Y Cf. A002117, A013661, A253905.

%Y Similar sequences: A368710, A368711, A368713.

%K nonn,easy

%O 1,4

%A _Amiram Eldar_, Jan 04 2024