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A379363 Numerators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

%I A379363 #10 Jan 08 2025 11:40:32
%S A379363 1,4,23,199,637,661,8953,9187,65869,201247,205927,26048,132697,134272,
%T A379363 135637,2190667,24424937,3513791,131554667,132348317,133227437,
%U A379363 938941259,947830139,190366027,2947643,74101331,223443593,2916305159,55809797621,55978686341,3437499844001
%N A379363 Numerators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).
%H A379363 Amiram Eldar, <a href="/A379363/b379363.txt">Table of n, a(n) for n = 1..1000</a>
%H A379363 László Tóth, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL13/Toth/toth10.html">A survey of gcd-sum functions</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.8.1. See pp. 18-19.
%H A379363 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 4.5, pp. 23-24.
%H A379363 Shiqin Chen and Wenguang Zhai, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Zhai/zhai4.html">Reciprocals of the Gcd-Sum Functions</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.8.3.
%F A379363 a(n) = numerator(Sum_{k=1..n} 1/A018804(k)).
%F A379363 a(n)/A379364(n) = Sum_{j=0..N} K_j/log(n)^(j-1/2) + O(1/log(n)^(N+1/2)), for any integer N >= 1, where K_j are constants, and in particular K_0 = (2/sqrt(Pi)) * Product_{p prime} (sqrt(1-1/p) * Sum_{k>=1} 1/A018804(p^k)) = 1.30088863073811791549... .
%e A379363 Fractions begin with 1, 4/3, 23/15, 199/120, 637/360, 661/360, 8953/4680, 9187/4680, 65869/32760, 201247/98280, 205927/98280, 26048/12285, ...
%t A379363 f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/pillai[n], {n, 1, 50}]]]
%o A379363 (PARI) pillai(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]);}
%o A379363 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / pillai(k); print1(numerator(s), ", "))};
%Y A379363 Cf. A018804, A272718, A370895, A379364 (denominators), A379365.
%K A379363 nonn,easy,frac
%O A379363 1,2
%A A379363 _Amiram Eldar_, Dec 21 2024