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A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).

%I A379517 #7 Dec 24 2024 07:27:22
%S A379517 1,2,5,17,37,43,15,109,225,239,1223,3809,1293,4019,1031,209,1693,1735,
%T A379517 5261,5345,5429,27649,306659,310619,312929,317549,4155857,4195897,
%U A379517 603091,615961,619393,19304143,19463731,1228951,9898103,4982299,1251116,2524397,10164083
%N A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).
%H A379517 Amiram Eldar, <a href="/A379517/b379517.txt">Table of n, a(n) for n = 1..1000</a>
%H A379517 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 A379517 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 A379517 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 A379517 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 A379517 a(n) = numerator(Sum_{k=1..n} 1/A047994(k)).
%F A379517 a(n)/A379518(n) = L * log(n) + M + O(log(n)^(5/3)/n), where L = A327837, M = L * (gamma - B + A1 + A2), gamma = A001620, B = Sum_{p prime} (1-1/p) * log(p) * Sum_{k>=1} k/(p^k*(p^k-1)) / A(p), A1 = Sum_{p prime} log(p)/(p^2*(p-1)*A(p)), A2 = Sum_{p prime} ((A*(p)(p)*log(p)/p^2), A(p) = 1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1)), and A*(p) = Sum_{k>=1} 1/(p^k*p^(k+1)-1)*A(p)) (Sita Ramaiah and Suryanarayana, 1980).
%e A379517 Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 109/28, 225/56, 239/56, 1223/280, 3809/840, ...
%t A379517 uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[1/uphi[n], {n, 1, 50}]]]
%o A379517 (PARI) uphi(n) = {my(f = factor(n)); prod(i = 1, #f~, -1 + f[i, 1]^f[i, 2]);}
%o A379517 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / uphi(k); print1(numerator(s), ", "))};
%Y A379517 Cf. A001620, A047994, A177754, A370899, A327837, A379518 (denominators), A379519.
%K A379517 nonn,easy,frac
%O A379517 1,2
%A A379517 _Amiram Eldar_, Dec 24 2024