This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A379583 #6 Dec 26 2024 20:02:58
%S A379583 1,2,3,13,17,21,25,51,467,539,611,629,701,773,845,1699,1843,1859,2003,
%T A379583 2039,2183,2327,2471,2489,62369,65969,198307,201007,211807,222607,
%U A379583 233407,467489,489089,510689,532289,532889,554489,576089,597689,600389,621989,643589,665189
%N A379583 Numerators of the partial sums of the reciprocals of the powerful part function (A057521).
%H A379583 Amiram Eldar, <a href="/A379583/b379583.txt">Table of n, a(n) for n = 1..1000</a>
%H A379583 Maurice-Étienne Cloutier, <a href="http://hdl.handle.net/20.500.11794/28374">Les parties k-puissante et k-libre d'un nombre</a>, Thèse de doctorat, Université Laval, Québec (2018).
%H A379583 Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Cloutier/cloutier2.html">On the powerful and squarefree parts of an integer</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
%H A379583 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.
%F A379583 a(n) = numerator(Sum_{k=1..n} 1/A057521(k)).
%F A379583 a(n)/A379584(n) = c * n + O(n^(1/2)), where c = A191622 (Cloutier et al., 2014). The error term was improved by Tóth (2017) to O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.
%e A379583 Fractions begin with 1, 2, 3, 13/4, 17/4, 21/4, 25/4, 51/8, 467/72, 539/72, 611/72, 629/72, ...
%t A379583 f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/powful[n], {n, 1, 50}]]]
%o A379583 (PARI) powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
%o A379583 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / powerful(k); print1(numerator(s), ", "))};
%Y A379583 Cf. A057521, A191622, A370902, A370903, A379584 (denominators), A379585.
%K A379583 nonn,easy,frac
%O A379583 1,2
%A A379583 _Amiram Eldar_, Dec 26 2024