A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).
1, 2, 11, 65, 79, 6, 55, 769, 10837, 30691, 33421, 32251, 34591, 16613, 34591, 1039561, 365327, 356647, 373573, 365513, 1504367, 4400261, 4569521, 4501817, 149447, 146327, 149603, 147263, 151631, 49937, 25651, 75913, 38639, 114097, 232289, 230129, 4470731, 4408487
Offset: 1
Examples
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, ...
Links
- Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.
- László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
Crossrefs
Programs
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Mathematica
Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[1, #] &, 60]]]
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PARI
lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(numerator(s), ", "))}; -
Python
from fractions import Fraction from sympy import divisor_sigma 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
Formula
a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/sigma(k)), where sigma(k) = A000203(k).
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).