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

1,2

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).

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

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

Sequence in context: A356552 A258567 A076396 * A076397 A254269 A264662

Adjacent sequences: A370831 A370832 A370833 * A370835 A370836 A370837

Keyword

nonn,easy

Author

Amiram Eldar, Mar 03 2024

Status

approved