A386921 Numerators of the partial sums of 1/d(prime(k)+1), where d is the number of divisors function.
1, 5, 13, 4, 3, 7, 23, 25, 53, 7, 5, 11, 23, 73, 377, 49, 67, 18, 113, 77, 41, 21, 257, 131, 68, 559, 287, 73, 599, 307, 629, 213, 109, 83, 337, 689, 719, 739, 1493, 1523, 4609, 4699, 33253, 34513, 34933, 35353, 36193, 36613, 37033, 37663, 38083, 7667, 7835, 7891
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
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Mathematica
Numerator[Accumulate[1/DivisorSigma[0, Prime[Range[100]] + 1]]]
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PARI
list(lim) = {my(s = 0); forprime(p = 1, lim, s += (1/numdiv(p+1)); print1(numerator(s), ", "));}
Formula
a(n) = numerator(Sum_{k=1..n} 1/A008329(k)).