A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).

1, 3, 3, 6, 6, 18, 18, 45, 90, 90, 90, 45, 45, 45, 15, 1, 3, 18, 18, 9, 9, 1, 3, 30, 15, 45, 90, 5, 15, 135, 135, 945, 945, 945, 945, 3780, 3780, 3780, 3780, 756, 756, 756, 756, 756, 756, 756, 756, 3780, 3780, 3780, 3780, 3780, 3780, 756, 756, 3780, 3780, 3780

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

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, A104529, A379357 (numerators).

Programs

Mathematica

f[p_, e_] := (e+1)*(e+2)/2; d3[1] = 1; d3[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[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(denominator(s), ", "))};

Formula

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