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A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.

%I A370834 #11 Mar 04 2024 01:37:08
%S A370834 1,2,2,3,2,5,3,2,3,7,2,3,3,5,3,11,5,2,3,13,7,5,3,5,3,2,3,17,3,7,19,7,
%T A370834 5,3,7,11,5,2,23,3,5,3,5,13,3,7,5,29,3,5,31,11,3,5,2,11,5,3,17,7,3,7,
%U A370834 11,13,37,7,19,13,7,5,41,3,7,5,43,11,3,5,5,2,23
%N A370834 a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.
%H A370834 Amiram Eldar, <a href="/A370834/b370834.txt">Table of n, a(n) for n = 1..10000</a>
%H A370834 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.
%H A370834 Michael De Vlieger, <a href="/A370834/a370834.png">Log log scatterplot of pi(a(n))</a>, n = 2..2^20, highlighting in red pi(a(n)) such that b(n) is a prime power, where b(n) = A001694(n). The remaining terms are such that b(n) is in A286708. The topmost red line corresponds with b(n) in A001248 (prime squares), the row of red dots at the bottom corresponds with b(n) in A000079 (powers of 2), the topmost blue line corresponds with b(n) > 16 in A069262 (4*p^2, with prime p).
%F A370834 a(n) = A006530(A001694(n)).
%F A370834 Sum_{A001694(n) <= x} a(n) = Sum_{i=1..k} e_i * x/log(x)^i + O(x/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = zeta(2)*zeta(3)/zeta(6) = 1.943596... (A082695) (De Koninck and Jakimczuk, 2024).
%t A370834 s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 1 &], f[[-1, 1]], Nothing]]; Array[s, 4000]
%o A370834 (PARI) lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 1, print1(f[#f~, 1], ", ")));}
%Y A370834 Cf. A001694, A006530, A073482, A082695, A370833, A370835.
%K A370834 nonn,easy
%O A370834 1,2
%A A370834 _Amiram Eldar_, Mar 03 2024