cp's OEIS Frontend

A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

%I A357845 #13 Oct 17 2022 01:43:26
%S A357845 1,2,11,65,79,6,55,769,10837,30691,33421,32251,34591,16613,34591,
%T A357845 1039561,365327,356647,373573,365513,1504367,4400261,4569521,4501817,
%U A357845 149447,146327,149603,147263,151631,49937,25651,75913,38639,114097,232289,230129,4470731,4408487
%N A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).
%H A357845 Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An alternating sum involving the reciprocal of certain multiplicative functions</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.
%H A357845 László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F A357845 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).
%F A357845 a(n)/A357846(n) ~ E * ((2/K-1)*(log(n) + gamma + F) + 2*log(2)*K'/K^2) + O(log(n)^(5/3)*log(log(n))^(4/3)/n), where E = Product_{p prime} alpha(p), F = Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p)), alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), K = A065442, K' = A065443 (Tóth, 2017).
%e A357845 Fractions begin with 1, 2/3, 11/12, 65/84, 79/84, 6/7, 55/56, 769/840, 10837/10920, 30691/32760, 33421/32760, 32251/32760, ...
%t A357845 Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]
%o A357845 (PARI) lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(numerator(s), ", "))};
%o A357845 (Python)
%o A357845 from fractions import Fraction
%o A357845 from sympy import divisor_sigma
%o A357845 def A357845(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1,n+1)).numerator # _Chai Wah Wu_, Oct 16 2022
%Y A357845 Cf. A000203, A065442, A065443, A068762, A357846 (denominators).
%Y A357845 Similar sequence: A104528, A212717, A357820.
%K A357845 nonn,frac
%O A357845 1,2
%A A357845 _Amiram Eldar_, Oct 16 2022