cp's OEIS Frontend

A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

%I A357843 #14 Oct 17 2022 01:43:18
%S A357843 1,1,1,2,7,11,17,7,3,5,7,19,25,11,25,113,143,133,163,51,14,51,61,117,
%T A357843 391,361,391,371,431,52,119,19,81,19,81,709,799,377,799,1553,1733,211,
%U A357843 467,226,467,889,979,961,1021,991,259,503,274,2147,2237,274,1141,274
%N A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).
%H A357843 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 A357843 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/d(k)), where d(k) = A000005(k).
%F A357843 a(n)/A357844(n) ~ n * Sum_{k=1..N} B_k/log(n)^(k-1/2) + O(n/log(n)^(N+1/2)), where B_k are constants, and in particular B_1 = (1/log(2) - 1) * (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Tóth, 2017).
%e A357843 Fractions begin with 1, 1/2, 1, 2/3, 7/6, 11/12, 17/12, 7/6, 3/2, 5/4, 7/4, 19/12, ...
%t A357843 Numerator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]
%o A357843 (PARI) lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(numerator(s), ", "))};
%o A357843 (Python)
%o A357843 from fractions import Fraction
%o A357843 from sympy import divisor_count
%o A357843 def A357843(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).numerator # _Chai Wah Wu_, Oct 16 2022
%Y A357843 Cf. A000005, A307704, A357844 (denominators).
%Y A357843 Similar sequences: A104528, A211177, A357820.
%K A357843 nonn,frac
%O A357843 1,4
%A A357843 _Amiram Eldar_, Oct 16 2022