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A368710 The maximal exponent in the prime factorization of the powerful numbers.

%I A368710 #15 Mar 29 2025 00:23:10
%S A368710 0,2,3,2,4,2,3,5,2,2,6,3,4,2,3,2,3,7,4,2,2,3,3,2,5,8,5,2,4,3,2,3,4,4,
%T A368710 2,2,3,9,2,6,4,4,3,2,6,4,5,2,5,2,2,3,5,3,10,2,3,7,2,2,4,3,3,3,2,3,2,2,
%U A368710 5,6,2,6,2,3,2,4,5,4,4,11,2,7,3,2,8,3,4
%N A368710 The maximal exponent in the prime factorization of the powerful numbers.
%H A368710 Amiram Eldar, <a href="/A368710/b368710.txt">Table of n, a(n) for n = 1..10000</a>
%H A368710 Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2018.8739">Generalizations of the Niven constant and the Feller-Tornier constant</a>, International Mathematical Forum, Vol. 13, No. 9 (2018), pp. 415-425.
%F A368710 a(n) = A051903(A001694(n)).
%F A368710 a(n) >= 2 for n >= 2.
%F A368710 Sum_{a(n)<=x} = D_{2,1} * sqrt(x) + O(sqrt(x)), where D_{2,1} = (6/Pi^2) * (2 + Sum_{k>=1} (A051903(k)+2)/(sqrt(k) * A048250(k))) (Jakimczuk, 2018; Theorem 2.1 and Remark 2.3).
%F A368710 Asymptotic mean (consequence of the formula above): Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = D_{2,1} * zeta(3)/zeta(3/2) = D_{2,1} / A090699.
%F A368710 The sum in the formula for D_{2,1} converges slowly: for k up to 10^8, 10^9 and 10^10 the sums are 14.845..., 14.908... and 14.938..., respectively. Thus, a lower bound for the value of this mean, calculated by summing over k=1..10^10, is 4.738... .
%t A368710 s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;;, 2]]] > 1 &]
%t A368710 (* or *)
%t A368710 f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1, 0, If[Min[e] > 1, Max[e], Nothing]]]; Array[f, 3000]
%o A368710 (PARI) lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1, print1(0, ", "), if(vecmin(e) > 1, print1(vecmax(e), ", "))));}
%Y A368710 Cf. A001694, A033150, A048250, A051903, A059956, A090699.
%Y A368710 Similar sequences: A368711, A368712, A368713.
%K A368710 nonn,easy
%O A368710 1,2
%A A368710 _Amiram Eldar_, Jan 04 2024