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A379619 Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

%I A379619 #11 Jan 08 2025 11:41:45
%S A379619 1,2,5,17,37,43,15,79,573,152,311,484,657,2041,4187,46897,94949,97589,
%T A379619 295847,300467,305087,310631,313151,63739,9181,9313,46961,47401,
%U A379619 333787,340717,68513,9863,49711,25103,6317,44549,89483,90253,181661,183047,9187,18605,18671
%N A379619 Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).
%H A379619 Amiram Eldar, <a href="/A379619/b379619.txt">Table of n, a(n) for n = 1..1000</a>
%H A379619 László Tóth, <a href="https://doi.org/10.2478/ausm-2014-0007">A survey of the alternating sum-of-divisors function</a>, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107. See p. 101, eq. (17).
%H A379619 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 4.14, p. 35.
%F A379619 a(n) = numerator(Sum_{k=1..n} 1/A206369(k)).
%F A379619 a(n)/A379620(n) = A * log(n) + B + O(n^(-1+eps)) for any eps > 0, where A and B are constants, A = Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/beta(p^k))) = 1.72360989673744398907... .
%e A379619 Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 79/20, 573/140, 152/35, 311/70, 484/105, ...
%t A379619 f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/beta[n], {n, 1, 50}]]]
%o A379619 (PARI) beta(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; sum(k = 0, e, (-1)^(e-k)*p^k)); }
%o A379619 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / beta(k); print1(numerator(s), ", "))};
%Y A379619 Cf. A206369, A370905, A370906, A379620 (denominators), A379621.
%K A379619 nonn,easy,frac
%O A379619 1,2
%A A379619 _Amiram Eldar_, Dec 27 2024