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

%I A379358 #7 Dec 22 2024 16:52:18
%S A379358 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,
%T A379358 90,5,15,135,135,945,945,945,945,3780,3780,3780,3780,756,756,756,756,
%U A379358 756,756,756,756,3780,3780,3780,3780,3780,3780,756,756,3780,3780,3780
%N A379358 Denominators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).
%D A379358 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.
%D A379358 József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.
%H A379358 Amiram Eldar, <a href="/A379358/b379358.txt">Table of n, a(n) for n = 1..10000</a>
%H A379358 Aleksandar Ivić, <a href="https://eudml.org/doc/259950">On the asymptotic formulae for some functions connected with powers of the zeta-function</a>, Matematički Vesnik, Vol. 1 (14) (29) (1977), pp. 79-90.
%F A379358 a(n) = denominator(Sum_{k=1..n} 1/A007425(k)).
%t A379358 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}]]]
%o A379358 (PARI) d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
%o A379358 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(denominator(s), ", "))};
%Y A379358 Cf. A007425, A061201, A104529, A379357 (numerators).
%K A379358 nonn,easy,frac
%O A379358 1,2
%A A379358 _Amiram Eldar_, Dec 21 2024