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A379367 Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

%I A379367 #6 Dec 22 2024 16:53:55
%S A379367 1,3,11,7,38,27,199,117,386,793,8933,1553,20574,41863,127591,71303,
%T A379367 1227166,2539417,48759433,24864701,25095646,50632187,1174239991,
%U A379367 605711068,125604071,252924241,267797099,19356010,564511331,1891973791,58959268151,31867258958,8730535499
%N A379367 Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).
%D A379367 Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.
%H A379367 Amiram Eldar, <a href="/A379367/b379367.txt">Table of n, a(n) for n = 1..1000</a>
%H A379367 N. G. de Bruijn, <a href="https://doi.org/10.1215/ijm/1255631814">On the number of integers <= x whose prime factors divide n</a>, Illinois Journal of Mathematics, Vol. 6, No. 1 (1962), pp. 137-141.
%H A379367 Olivier Robert and Gérald Tenenbaum, <a href="https://doi.org/10.1016/j.indag.2013.07.007">Sur la répartition du noyau d'un entier</a>, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 802-914.
%H A379367 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 4.6, pp. 24-26.
%F A379367 a(n) = numerator(Sum_{k=1..n} 1/A007947(k)).
%F A379367 a(n)/A379368(n) = exp((1 + o(1)) * sqrt(8*log(n)/log(log(n)))).
%F A379367 a(n)/A379368(n) ~ (1/2) * exp(gamma) * F(log(n)) * log(log(n)), where F(t) = (6/Pi^2) * Sum_{m>=1} min(1,exp(t)/m)/Product_{primes p|m} (p+1).
%e A379367 Fractions begin with 1, 3/2, 11/6, 7/3, 38/15, 27/10, 199/70, 117/35, 386/105, 793/210, 8933/2310, 1553/385, ...
%t A379367 rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]
%o A379367 (PARI) rad(n) = vecprod(factor(n)[, 1]);
%o A379367 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(numerator(s), ", "))};
%Y A379367 Cf. A007947, A073355, A370896, A379368 (denominators), A379369.
%Y A379367 Cf. A059956, A073004.
%K A379367 nonn,easy,frac
%O A379367 1,2
%A A379367 _Amiram Eldar_, Dec 21 2024