cp's OEIS Frontend

A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

%I A386921 #8 Aug 09 2025 07:47:26
%S A386921 1,5,13,4,3,7,23,25,53,7,5,11,23,73,377,49,67,18,113,77,41,21,257,131,
%T A386921 68,559,287,73,599,307,629,213,109,83,337,689,719,739,1493,1523,4609,
%U A386921 4699,33253,34513,34933,35353,36193,36613,37033,37663,38083,7667,7835,7891
%N A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.
%H A386921 Amiram Eldar, <a href="/A386921/b386921.txt">Table of n, a(n) for n = 1..10000</a>
%H A386921 Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, <a href="https://arxiv.org/abs/2304.04805">Karatsuba's divisor problem and related questions</a>, arXiv:2304.04805 [math.NT], 2023.
%H A386921 Vitalii V. Iudelevich, <a href="https://doi.org/10.4213/im9270e">On the Karatsuba divisor problem</a>, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; <a href="https://arxiv.org/abs/2304.03049">arXiv preprint</a>, arXiv:2304.03049 [math.NT], 2023.
%F A386921 a(n) = numerator(Sum_{k=1..n} 1/A008329(k)).
%F A386921 a(n)/A386922(n) <= 4 * K * n/log(n)^(3/2) + O(n*log(log(n))/log(n)^(5/2)), where K = (1/sqrt(Pi)) * Product_{p prime} sqrt(p/(p-1)) * (p * log(p/(p-1)) - 1/(p-1)) = 0.25320111501639846923... (Iudelevich, 2022). Gabdullin et al. (2023) conjectured that a(n)/A386922(n) ~ K * n/log(n)^(3/2).
%e A386921 Fractions begin with 1/2, 5/6, 13/12, 4/3, 3/2, 7/4, 23/12, 25/12, 53/24, 7/3, 5/2, 11/4, ...
%t A386921 Numerator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]
%o A386921 (PARI) list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(numerator(s), ", "));}
%Y A386921 Cf. A000005, A008329, A008864, A104528, A386922 (denominators).
%K A386921 nonn,frac,easy
%O A386921 1,2
%A A386921 _Amiram Eldar_, Aug 08 2025