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A368710
The maximal exponent in the prime factorization of the powerful numbers.
7
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, 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, 5, 6, 2, 6, 2, 3, 2, 4, 5, 4, 4, 11, 2, 7, 3, 2, 8, 3, 4
OFFSET
1,2
LINKS
Rafael Jakimczuk, Generalizations of the Niven constant and the Feller-Tornier constant, International Mathematical Forum, Vol. 13, No. 9 (2018), pp. 415-425.
FORMULA
a(n) = A051903(A001694(n)).
a(n) >= 2 for n >= 2.
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).
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.
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... .
MATHEMATICA
s[n_] := If[n == 1, 0, Max @@ Last /@ FactorInteger[n]]; s /@ Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]
(* or *)
f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[n == 1, 0, If[Min[e] > 1, Max[e], Nothing]]]; Array[f, 3000]
PROG
(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), ", ")))); }
CROSSREFS
Similar sequences: A368711, A368712, A368713.
Sequence in context: A183152 A210942 A368925 * A080771 A025477 A080189
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Jan 04 2024
STATUS
approved