A174172 Partials sums of A001694.

1, 5, 13, 22, 38, 63, 90, 122, 158, 207, 271, 343, 424, 524, 632, 753, 878, 1006, 1150, 1319, 1515, 1715, 1931, 2156, 2399, 2655, 2943, 3232, 3556, 3899, 4260, 4652, 5052, 5484, 5925, 6409, 6909, 7421, 7950, 8526, 9151, 9799, 10474, 11150, 11879, 12663

Offset: 1

Author

Jonathan Vos Post, Mar 10 2010

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk, The kernel of powerful numbers, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Theorem 2.7, p. 729.

Crossrefs

Cf. A001694, A090699.

Programs

Mathematica

Accumulate @ Select[Range[1000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &] (* Amiram Eldar, Jan 30 2023 *)

PARI

lista(kmax) = {my(s = 0); for(k = 1, kmax, if(k==1 || vecmin(factor(k)[, 2]) > 1, s += k; print1(s, ", ")));} \\ Amiram Eldar, May 13 2023

Formula

a(n) = Sum_{i=1..n} A001694(i).

a(n) ~ (zeta(3)^2/(3*zeta(3/2)^2)) * n^3. - Amiram Eldar, Jan 30 2023

a(n) = c * A001694(n)^(3/2) + o(A001694(n)^(3/2)), where c = zeta(3/2)/(3*zeta(3)) = 0.7244181041... (Jakimczuk, 2017). - Amiram Eldar, May 13 2023