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A370833 a(n) is the greatest prime dividing the n-th cubefree number, for n >= 2; a(1)=1.

%I A370833 #15 Aug 06 2024 02:05:21
%S A370833 1,2,3,2,5,3,7,3,5,11,3,13,7,5,17,3,19,5,7,11,23,5,13,7,29,5,31,11,17,
%T A370833 7,3,37,19,13,41,7,43,11,5,23,47,7,5,17,13,53,11,19,29,59,5,61,31,7,
%U A370833 13,11,67,17,23,7,71,73,37,5,19,11,13,79,41,83,7,17,43
%N A370833 a(n) is the greatest prime dividing the n-th cubefree number, for n >= 2; a(1)=1.
%H A370833 Amiram Eldar, <a href="/A370833/b370833.txt">Table of n, a(n) for n = 1..10000</a>
%H A370833 Jean-Marie De Koninck and Rafael Jakimczuk, <a href="https://doi.org/10.33044/revuma.3154">Summing the largest prime factor over integer sequences</a>, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.
%F A370833 a(n) = A006530(A004709(n)).
%F A370833 Sum_{A004709(n) <= x} a(n) = Sum_{i=1..k} d_i * x^2/log(x)^i + O(x^2/log(x)^(k+1)), for any given positive integer k, where d_i are constants, d_1 = 315/(4*Pi^4) = 0.808446... (De Koninck and Jakimczuk, 2024).
%t A370833 s[n_] := Module[{f = FactorInteger[n]}, If[AllTrue[f[[;; , 2]], # < 3 &], f[[-1, 1]], Nothing]]; Array[s, 200]
%o A370833 (PARI) lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmax(f[, 2]) < 3, print1(f[#f~, 1], ", ")));}
%o A370833 (Python)
%o A370833 from sympy import mobius, integer_nthroot, primefactors
%o A370833 def A370833(n):
%o A370833     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
%o A370833     m, k = n, f(n)
%o A370833     while m != k:
%o A370833         m, k = k, f(k)
%o A370833     return max(primefactors(m),default=1) # _Chai Wah Wu_, Aug 06 2024
%Y A370833 Cf. A004709, A006530, A073482, A370834, A370835.
%K A370833 nonn,easy
%O A370833 1,2
%A A370833 _Amiram Eldar_, Mar 03 2024