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A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.

%I A386922 #7 Aug 09 2025 07:47:04
%S A386922 2,6,12,3,2,4,12,12,24,3,2,4,8,24,120,15,20,5,30,20,10,5,60,30,15,120,
%T A386922 60,15,120,60,120,40,20,15,60,120,120,120,240,240,720,720,5040,5040,
%U A386922 5040,5040,5040,5040,5040,5040,5040,1008,1008,1008,1008,63,1008,5040,5040
%N A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.
%H A386922 Amiram Eldar, <a href="/A386922/b386922.txt">Table of n, a(n) for n = 1..10000</a>
%H A386922 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 A386922 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 A386922 a(n) = denominator(Sum_{k=1..n} 1/A008329(k)).
%e A386922 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 A386922 Denominator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]
%o A386922 (PARI) list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(denominator(s), ", "));}
%Y A386922 Cf. A000005, A008329, A008864, A104529, A386921 (numerators).
%K A386922 nonn,frac,easy
%O A386922 1,1
%A A386922 _Amiram Eldar_, Aug 08 2025