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A379364 Denominators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).

%I A379364 #8 Jan 08 2025 11:40:42
%S A379364 1,3,15,120,360,360,4680,4680,32760,98280,98280,12285,61425,61425,
%T A379364 61425,982800,10810800,1544400,57142800,57142800,57142800,399999600,
%U A379364 399999600,79999920,1230768,30769200,92307600,1199998800,22799977200,22799977200,1390798609200,695399304600
%N A379364 Denominators of the partial sums of the reciprocals of Pillai's arithmetical function (A018804).
%H A379364 Amiram Eldar, <a href="/A379364/b379364.txt">Table of n, a(n) for n = 1..1000</a>
%H A379364 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 A379364 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 A379364 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 A379364 a(n) = denominator(Sum_{k=1..n} 1/A018804(k)).
%t A379364 f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[1/pillai[n], {n, 1, 50}]]]
%o A379364 (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 A379364 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / pillai(k); print1(denominator(s), ", "))};
%Y A379364 Cf. A018804, A272718, A370895, A379363 (numerators), A379366.
%K A379364 nonn,easy,frac
%O A379364 1,2
%A A379364 _Amiram Eldar_, Dec 21 2024