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A370834
a(n) is the greatest prime dividing the n-th powerful number, for n >= 2; a(1)=1.
3
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, 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, 11, 13, 37, 7, 19, 13, 7, 5, 41, 3, 7, 5, 43, 11, 3, 5, 5, 2, 23
OFFSET
1,2
LINKS
Jean-Marie De Koninck and Rafael Jakimczuk, Summing the largest prime factor over integer sequences, Revista de la Unión Matemática Argentina, Vol. 67, No. 1 (2024), pp. 27-35.
Michael De Vlieger, Log log scatterplot of pi(a(n)), 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).
FORMULA
a(n) = A006530(A001694(n)).
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).
MATHEMATICA
s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 1 &], f[[-1, 1]], Nothing]]; Array[s, 4000]
PROG
(PARI) lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 1, print1(f[#f~, 1], ", "))); }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 03 2024
STATUS
approved