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

%I A379357 #8 Dec 22 2024 16:52:05
%S A379357 1,4,5,11,13,41,47,122,259,269,299,152,167,172,59,4,13,79,85,43,44,5,
%T A379357 16,161,83,254,517,29,92,833,878,6191,6296,6401,6506,26129,27389,
%U A379357 27809,28229,5671,5923,5951,6203,6245,6287,6371,6623,33199,33829,34039,34459,34669
%N A379357 Numerators of the partial sums of the reciprocals of the 3rd Piltz function d_3(n) (A007425).
%D A379357 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 A379357 József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter II, page 59.
%H A379357 Amiram Eldar, <a href="/A379357/b379357.txt">Table of n, a(n) for n = 1..10000</a>
%H A379357 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 A379357 a(n) = numerator(Sum_{k=1..n} 1/A007425(k)).
%F A379357 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.
%e A379357 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, ...
%t A379357 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}]]]
%o A379357 (PARI) d3(n) = vecprod(apply(e -> (e+1)*(e+2)/2, factor(n)[, 2]));
%o A379357 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / d3(k); print1(numerator(s), ", "))};
%Y A379357 Cf. A007425, A061201, A104528, A379358 (denominators).
%K A379357 nonn,easy,frac
%O A379357 1,2
%A A379357 _Amiram Eldar_, Dec 21 2024