A386925 a(n) = numerator(Sum_{k=1..n} d(k+1)/d(k)), where d is the number of divisors function.

2, 3, 9, 31, 43, 23, 29, 125, 47, 49, 61, 187, 211, 223, 119, 607, 697, 707, 797, 817, 847, 431, 491, 3973, 4133, 4253, 4433, 1491, 1651, 1661, 1781, 5423, 5543, 5663, 5933, 17879, 18599, 18959, 19679, 19769, 21209, 21299, 22379, 22739, 22979, 23159, 24959, 25067

Offset: 1

Author

Amiram Eldar, Aug 08 2025

Keywords

• nonn
• frac
• easy

Examples

			Fractions begin with 2, 3, 9/2, 31/6, 43/6, 23/3, 29/3, 125/12, 47/4, 49/4, 61/4, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Florian Luca and Igor E. Shparlinski, On the values of the divisor function, Monatshefte für Mathematik, Vol. 154, No. 1 (2008), pp. 59-69.

Crossrefs

Cf. A000005, A386926 (denominators).

Programs

Mathematica

With[{s = DivisorSigma[0, Range[100]]}, Numerator[Accumulate[Rest[s]/Most[s]]]]

PARI

list(nmax) = {my(s = 0, d1 = 1, d2); for(n = 2, nmax, d2 = numdiv(n); s += (d2/d1); print1(numerator(s), ", "); d1 = d2);}

Formula

a(n)/A386926(n) ≍ n * sqrt(log(n)) (Luca and Shparlinski, 2008).