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%I #57 Feb 23 2024 07:27:00
%S 0,1,1,1,1,5,1,3,2,7,1,4,1,9,8,2,1,7,1,6,10,13,1,11,2,15,1,8,1,31,1,5,
%T 14,19,12,5,1,21,16,17,1,41,1,12,13,25,1,7,2,9,20,14,1,3,16,23,22,31,
%U 1,23,1,33,17,3,18,61,1,18,26,59,1,13,1,39,11,20,18,71,1,11,4,43,1,31,22
%N Numerator of r(n) = Sum(e/p: n=Product(p^e)).
%C Least common multiple of n and its arithmetic derivative, divided by n, i.e. a(n) = lcm(n,n')/n = A086130(n)/A000027(n). - _Giorgio Balzarotti_, Apr 14 2011
%H Antti Karttunen, <a href="/A083345/b083345.txt">Table of n, a(n) for n = 1..65537</a>
%F The fraction a(n)/A083346(n) is totally additive with a(p) = 1/p. - _Franklin T. Adams-Watters_, May 17 2006
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A083346(k) = Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141). - _Amiram Eldar_, Sep 29 2023
%F a(n) = A003415(n) / A085731(n) = A342001(n) / A369008(n). - _Antti Karttunen_, Jan 16 2024
%e Fractions begin with 0, 1/2, 1/3, 1, 1/5, 5/6, 1/7, 3/2, 2/3, 7/10, 1/11, 4/3, ...
%e For n = 12, 2*2*3 = 2^2 * 3^1 --> r(12) = 2/2 + 1/3 = (6+2)/6, therefore a(12) = 4, A083346(12) = 3.
%e For n = 18, 2*3*3 = 2^1 * 3^2 --> r(18) = 1/2 + 2/3 = (3+4)/6, therefore a(18) = 7, A083346(18) = 6.
%t Array[Numerator@ Total[FactorInteger[#] /. {p_, e_} /; e > 0 :> e/p] - Boole[# == 1] &, 85] (* _Michael De Vlieger_, Feb 25 2018 *)
%o (PARI) A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~,i,f[i,2]/f[i,1]))); }; \\ _Antti Karttunen_, Feb 25 2018
%Y Cf. A083346 (denominator), A000027, A072873, A083347, A083348, A085731, A086130, A136141, A342001, A342002 [= a(A276086(n))].
%Y Cf. A369002 (positions of even terms), A369003 (of odd terms), A369005 (of multiples of 4), A369007 (of multiples of 27), A369008, A369068 (Möbius transform), A369069.
%K nonn,easy,frac
%O 1,6
%A _Reinhard Zumkeller_, Apr 25 2003
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