A386923 a(n) = numerator(Sum_{k=1..n} d(k)/d(k+1)), where d is the number of divisors function.

1, 3, 13, 11, 25, 37, 20, 8, 35, 43, 133, 169, 175, 187, 983, 1133, 1153, 1333, 451, 481, 501, 541, 273, 899, 1843, 1903, 1943, 2123, 1069, 1189, 1199, 622, 637, 652, 1976, 4357, 2201, 2246, 4537, 4897, 9839, 10559, 10619, 10799, 11069, 11429, 2293, 2413, 2431

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Maxim A. Korolev, On Karatsuba's problem concerning the divisor function, Monatshefte für Mathematik, Vol. 168, No. 3 (2012), pp. 403-441; arXiv preprint, arXiv:1011.1391 [math.NT], 2010.

Formula

a(n)/A386924(n) = K * n * sqrt(log(n)) + O(n*log(log(n))), where K = (1/sqrt(Pi)) * Product_{p prime} (1/sqrt(p*(p-1)) + sqrt(1-1/p) * (p-1) * log(p/(p-1))) = 0.75782771069999213465... (Korolev, 2012).

Example

Fractions begin with 1/2, 3/2, 13/6, 11/3, 25/6, 37/6, 20/3, 8, 35/4, 43/4, 133/12, ...

Mathematica

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

Prog

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

Crossrefs

Cf. A000005, A386924 (denominators).

Sequence in context: A085416 A107802 A142351 * A272837 A273576 A272849

Adjacent sequences: A386920 A386921 A386922 * A386924 A386925 A386926

Keyword

nonn,frac,easy

Author

Amiram Eldar, Aug 08 2025

Status

approved