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A378680 a(n) = numerator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

%I A378680 #8 Dec 05 2024 09:27:16
%S A378680 1,1,3,11,7,17,10,11,25,9,5,16,35,19,98,211,221,118,41,87,271,143,146,
%T A378680 151,317,109,57,176,367,377,196,407,2879,2921,997,516,1583,1604,3313,
%U A378680 3383,1744,593,1221,3733,1919,388,395,811,275,1389,4237,2171,2192,4489
%N A378680 a(n) = numerator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.
%H A378680 Amiram Eldar, <a href="/A378680/b378680.txt">Table of n, a(n) for n = 1..1000</a>
%H A378680 Jean-Marie De Koninck, <a href="https://doi.org/10.1007/BF01388417">Sur les plus grands facteurs premiers d'un entier</a>, Monatshefte für Mathematik, Vol. 116, No. 1 (1993), pp. 13-37; <a href="https://eudml.org/doc/178614">alternative link</a>; <a href="https://www.jeanmariedekoninck.mat.ulaval.ca/fileadmin/Documents/Publications/1993_sur_les_plus_grands_facteurs_premiers_d_un_entier.pdf">author's copy</a>.
%F A378680 a(n)/A378681(n) = Sum_{k=1..m} c_k * n/log(n)^k + O(n/log(n)^(m+1)) for any integer m >= 1, where c_k are constants. c_1 = Sum_{k>=1} (1/k)*Sum_{p prime > P(k)} 1/p^2 = Sum_{p prime} (1/p^2)*Product_{primes q < p} (1/(1-1/q)) = 1.254435359..., where P(k) = A006530(k) is the greatest prime dividing k for k >= 2, and P(1) = 1.
%e A378680 Fractions begin: 1/2, 1, 3/2, 11/6, 7/3, 17/6, 10/3, 11/3, 25/6, 9/2, 5, 16/3, ...
%t A378680 p2[c_] := Module[{f = FactorInteger[c]}, If[f[[-1, 2]] > 1, f[[-1, 1]], f[[-2, 1]]]]; Numerator@ Accumulate[Table[1/p2[c], {c, Select[Range[100], CompositeQ]}]]
%o A378680 (PARI) lista(nmax) = {my(s = 0); forcomposite(n = 1, nmax, f = factor(n); s += if(f[#f~, 2] > 1, 1/f[#f~, 1], 1/f[#f~ - 1, 1]); print1(numerator(s), ", "));}
%Y A378680 Cf. A006530, A087039, A087040, A378681 (denominators).
%K A378680 nonn,easy,frac
%O A378680 1,3
%A A378680 _Amiram Eldar_, Dec 03 2024