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%I #8 Dec 06 2024 11:29:20
%S 1,3,11,7,38,43,316,737,269,283,3183,10319,136457,140747,146753,80884,
%T 1390043,1475128,28282687,29252656,29945491,30386386,703736723,
%U 246972956,50881877,51453910,161798159,164985200,4806880087,24681369758,768357309113,1636994863291
%N a(n) = numerator(Sum_{k=1..n} 1/P(k)), where P(k) = A006530(k) is the greatest prime dividing k for k >= 2, and P(1) = 1.
%D József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory, Vol. 1, Springer, 2006, Chapter IV, p. 122.
%H Amiram Eldar, <a href="/A378678/b378678.txt">Table of n, a(n) for n = 1..1000</a>
%H Paul Erdős and Aleksandar Ivić, <a href="https://users.renyi.hu/~p_erdos/1980-08.pdf">Estimates for sums involving the largest prime factor of an integer and certain related additive functions</a>, Studia Sci. Math. Hungar, Vol. 15, No. 1-3 (1980), pp. 183-199.
%H Paul Erdős and Aleksandar Ivić, <a href="https://eudml.org/doc/255308">On sums involving reciprocals of certain arithmetical functions</a>, Publ. Inst. Math.(Beograd)(NS), Vol. 32, No. 46 (1982), pp. 49-56.
%H Paul Erdős, Aleksandar Ivić, and Carl Pomerance, <a href="https://static.renyi.hu/~p_erdos/1986-10.pdf">On sums involving reciprocals of the largest prime factor of an integer</a>, Glasnik Matematicki, Vol. 21, No. 41 (1986), pp. 283-300.
%H Aleksandar Ivić, <a href="https://doi.org/10.1007/BF01223669">Sum of reciprocals of the largest prime factor of an integer</a>, Archiv der Mathematik, Vol. 36 (1981), pp. 57-61.
%H Aleksandar Ivić, <a href="https://eudml.org/doc/206771">On sums involving reciprocals of the largest prime factor of an integer II</a>, Acta Arithmetica, Vol. 71, No. 3 (1995), pp. 229-251
%H Aleksandar Ivić and Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/paper48.pdf">Estimates for certain sums involving the largest prime factor of an integer</a>, Coll. Math. Soc. J. Bolyai, Vol. 34 (1981), pp. 769-789.
%F a(n)/A378679(n) = n * exp(-sqrt(2*log(n)*log(log(n))) + O(sqrt(log(n)*log(log(log(n)))))) (Ivić, 1981).
%e Fractions begin: 1, 3/2, 11/6, 7/3, 38/15, 43/15, 316/105, 737/210, 269/70, 283/70, 3183/770, 10319/2310, ...
%t Numerator@ Accumulate[Table[1/FactorInteger[n][[-1, 1]], {n, 1, 33}]]
%o (PARI) lista(nmax) = {my(s = 1); print1(1, ", "); for(n = 2, nmax, f = factor(n); s += 1/f[#f~, 1]; print1(numerator(s), ", "));}
%Y Cf. A006530, A057158, A378679 (denominators).
%K nonn,easy,frac,new
%O 1,2
%A _Amiram Eldar_, Dec 03 2024