cp's OEIS Frontend

A379585 Numerators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

%I A379585 #6 Dec 26 2024 20:03:13
%S A379585 1,0,1,3,7,3,7,13,125,53,125,107,179,107,179,349,493,53,69,65,81,65,
%T A379585 81,79,1991,1591,43357,40657,51457,40657,51457,102239,123839,102239,
%U A379585 123839,123239,144839,123239,144839,142139,163739,142139,163739,158339,160739,139139
%N A379585 Numerators of the partial alternating sums of the reciprocals of the powerful part function (A057521).
%H A379585 Amiram Eldar, <a href="/A379585/b379585.txt">Table of n, a(n) for n = 1..1000</a>
%H A379585 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.12, p. 33.
%F A379585 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A057521(k)).
%F A379585 a(n)/A379586(n) = (5/19) * A191622 * n + O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and with an improved error term O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.
%e A379585 Fractions begin with 1, 0, 1, 3/4, 7/4, 3/4, 7/4, 13/8, 125/72, 53/72, 125/72, 107/72, ...
%t A379585 f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/powful[n], {n, 1, 50}]]]
%o A379585 (PARI) powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
%o A379585 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powerful(k); print1(numerator(s), ", "))};
%Y A379585 Cf. A057521, A191622, A370902, A370903, A379583, A379586 (denominators).
%K A379585 nonn,easy,frac
%O A379585 1,4
%A A379585 _Amiram Eldar_, Dec 26 2024