A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

1, 4, 5, 11, 13, 41, 47, 122, 259, 269, 299, 152, 167, 172, 59, 4, 13, 79, 85, 43, 44, 5, 16, 161, 83, 254, 517, 29, 92, 833, 878, 6191, 6296, 6401, 6506, 26129, 27389, 27809, 28229, 5671, 5923, 5951, 6203, 6245, 6287, 6371, 6623, 33199, 33829, 34039, 34459, 34669

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 4/3, 5/3, 11/6, 13/6, 41/18, 47/18, 122/45, 259/90, 269/90, 299/90, 152/45, ...
		

References

• Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 12-13, Theorem 1.2.
• József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Aleksandar Ivić, On the asymptotic formulae for some functions connected with powers of the zeta-function, Matematički Vesnik, Vol. 1 (14) (29) (1977), pp. 79-90.

Crossrefs

Cf. A007425, A061201, A104528, A379358 (denominators).

Programs

Mathematica

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/d3[n], {n, 1, 100}]]]

PARI

d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} 1/A007425(k)).

a(n)/A379358(n) = Sum_{i=1..N} b_i * n / log(n)^(i-1/3) + O(n / log(n)^(N+1-1/3)), for any fixed N >= 1, where b_i are constants. The same formula holds (with different constants) for any Piltz function d_k(n), for k >= 2, when 1/3 is replaced by 1/k.