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A379518 Denominators of the partial sums of the reciprocals of the unitary totient function (A047994).

%I A379518 #7 Dec 24 2024 07:26:46
%S A379518 1,1,2,6,12,12,4,28,56,56,280,840,280,840,210,42,336,336,1008,1008,
%T A379518 1008,5040,55440,55440,55440,55440,720720,720720,102960,102960,102960,
%U A379518 3191760,3191760,199485,1595880,797940,199485,398970,1595880,11171160,1117116,279279
%N A379518 Denominators of the partial sums of the reciprocals of the unitary totient function (A047994).
%H A379518 Amiram Eldar, <a href="/A379518/b379518.txt">Table of n, a(n) for n = 1..1000</a>
%H A379518 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 52.
%H A379518 V. Sita Ramaiah and D. Suryanarayana, <a href="https://web.archive.org/web/20200803214209/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005bab_1334.pdf">Sums of reciprocals of some multiplicative functions - II</a>, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
%H A379518 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 4.10, pp. 30-31.
%H A379518 Rimer Zurita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zurita/zur3.html">Generalized Alternating Sums of Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.5, pp. 16-17.
%F A379518 a(n) = denominator(Sum_{k=1..n} 1/A047994(k)).
%t A379518 uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Denominator[Accumulate[Table[1/uphi[n], {n, 1, 50}]]]
%o A379518 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}
%o A379518 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / uphi(k); print1(denominator(s), ", "))};
%Y A379518 Cf. A047994, A177754, A370899, A379517 (numerators), A379520.
%K A379518 nonn,easy,frac
%O A379518 1,3
%A A379518 _Amiram Eldar_, Dec 24 2024