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A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).

%I A379514 #7 Dec 24 2024 07:28:42
%S A379514 1,3,12,60,20,30,120,360,360,40,120,120,840,70,840,14280,42840,42840,
%T A379514 42840,42840,171360,57120,57120,171360,2227680,2227680,2227680,
%U A379514 2227680,2227680,247520,123760,4084080,340340,9189180,36756720,183783600,3491888400,3491888400
%N A379514 Denominators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).
%H A379514 Amiram Eldar, <a href="/A379514/b379514.txt">Table of n, a(n) for n = 1..1000</a>
%H A379514 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. 51.
%H A379514 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 A379514 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 4.9, pp. 28-29.
%H A379514 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 4.3, pp. 12-15.
%F A379514 a(n) = denominator(Sum_{k=1..n} 1/A034448(k)).
%t A379514 usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]
%o A379514 (PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
%o A379514 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(denominator(s), ", "))};
%Y A379514 Cf. A034448, A064609, A370898, A379513 (numerators), A379516.
%K A379514 nonn,easy,frac
%O A379514 1,2
%A A379514 _Amiram Eldar_, Dec 23 2024