A386922 Denominators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.
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, 60, 15, 120, 60, 120, 40, 20, 15, 60, 120, 120, 120, 240, 240, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 1008, 1008, 1008, 1008, 63, 1008, 5040, 5040
Offset: 1
Examples
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, ...
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Mikhail R. Gabdullin, Vitalii V. Iudelevich, and Sergei V. Konyagin, Karatsuba's divisor problem and related questions, arXiv:2304.04805 [math.NT], 2023.
- Vitalii V. Iudelevich, On the Karatsuba divisor problem, Izvestiya: Mathematics, Vol. 86, No. 5 (2022), pp. 992-1019; arXiv preprint, arXiv:2304.03049 [math.NT], 2023.
Programs
-
Mathematica
Denominator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]
-
PARI
list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(denominator(s), ", "));}
Formula
a(n) = denominator(Sum_{k=1..n} 1/A008329(k)).