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A379586 Denominators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

%I A379586 #6 Dec 26 2024 20:03:20
%S A379586 1,1,1,4,4,4,4,8,72,72,72,72,72,72,72,144,144,16,16,16,16,16,16,16,
%T A379586 400,400,10800,10800,10800,10800,10800,21600,21600,21600,21600,21600,
%U A379586 21600,21600,21600,21600,21600,21600,21600,21600,21600,21600,21600,21600,1058400
%N A379586 Denominators of the partial alternating sums of the reciprocals of the powerful part function (A057521).
%H A379586 Amiram Eldar, <a href="/A379586/b379586.txt">Table of n, a(n) for n = 1..1000</a>
%H A379586 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 A379586 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A057521(k)).
%t A379586 f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/powful[n], {n, 1, 50}]]]
%o A379586 (PARI) powerful(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] > 1, f[i, 1]^f[i, 2], 1)); }
%o A379586 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powerful(k); print1(denominator(s), ", "))};
%Y A379586 Cf. A057521, A370902, A370903, A379584, A379585 (numerators).
%K A379586 nonn,easy,frac
%O A379586 1,4
%A A379586 _Amiram Eldar_, Dec 26 2024