A370835 a(n) is the greatest prime dividing the n-th cubefull number, for n >= 2; a(1)=1.

1, 2, 2, 3, 2, 2, 3, 5, 2, 3, 3, 2, 7, 3, 2, 5, 3, 3, 3, 5, 2, 3, 11, 3, 3, 5, 2, 3, 13, 7, 3, 7, 5, 5, 3, 3, 5, 2, 17, 5, 3, 7, 3, 3, 19, 3, 3, 5, 2, 7, 5, 5, 3, 11, 7, 3, 23, 3, 11, 3, 5, 5, 2, 7, 5, 3, 13, 7, 3, 5, 3, 11, 7, 3, 29, 5, 5, 3, 7, 13, 31, 5, 3, 5

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.

Crossrefs

Cf. A036966, A006530, A073482, A082695, A370833, A370834.

Programs

Mathematica

s[n_] := Module[{f = FactorInteger[n]}, If[n == 1 || AllTrue[f[[;; , 2]], # > 2 &], f[[-1, 1]], Nothing]]; Array[s, 32000]

PARI

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

Formula

a(n) = A006530(A036966(n)).

Sum_{A036966(n) <= x} a(n) = Sum_{i=1..k} e_i * x^(2/3)/log(x)^i + O(x^(2/3)/log(x)^(k+1)), for any given positive integer k, where e_i are constants, e_1 = (3/2) * Product_{p prime} (1 + Sum_{i>=3} 1/p^(2*i/3)) = 3.44968588450293915243... (De Koninck and Jakimczuk, 2024).