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Partials sums of A001694.
2

%I #23 May 13 2023 06:16:51

%S 1,5,13,22,38,63,90,122,158,207,271,343,424,524,632,753,878,1006,1150,

%T 1319,1515,1715,1931,2156,2399,2655,2943,3232,3556,3899,4260,4652,

%U 5052,5484,5925,6409,6909,7421,7950,8526,9151,9799,10474,11150,11879,12663

%N Partials sums of A001694.

%H Amiram Eldar, <a href="/A174172/b174172.txt">Table of n, a(n) for n = 1..10000</a>

%H Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7759">The kernel of powerful numbers</a>, International Mathematical Forum, Vol. 12, No. 15 (2017), pp. 721-730, Theorem 2.7, p. 729.

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

%F a(n) ~ (zeta(3)^2/(3*zeta(3/2)^2)) * n^3. - _Amiram Eldar_, Jan 30 2023

%F 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

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

%o (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

%Y Cf. A001694, A090699.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Mar 10 2010