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

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

Author

Amiram Eldar, Mar 03 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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).

Crossrefs

Cf. A001694, A006530, A073482, A082695, A370833, A370835.

Programs

Mathematica

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 1 &], f[[-1, 1]], Nothing]]; Array[s, 4000]

PARI

lista(kmax) = {my(f); print1(1, ", "); for(k = 2, kmax, f = factor(k); if(vecmin(f[, 2]) > 1, print1(f[#f~, 1], ", ")));}

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).