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A379581 Numerators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

%I A379581 #8 Dec 26 2024 20:02:40
%S A379581 1,1,5,-1,1,-2,1,-104,1,-19,1,-769,-7687,-4916,-261,-1262,-20453,
%T A379581 -57923,-1066503,-5979161,-17475593,-8958244,-201189767,-79457304,
%U A379581 -42275159,-87410483,-13046193,-23669663,-612055937,-1025912126,-28568429291,-128848674356,-125809879051
%N A379581 Numerators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).
%H A379581 Amiram Eldar, <a href="/A379581/b379581.txt">Table of n, a(n) for n = 1..1000</a>
%H A379581 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 4.11, pp. 31-32.
%F A379581 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A055231(k)).
%F A379581 a(n)/A379582(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = ((9-12*sqrt(2))/23) * A328013, and B = ((2^(5/3) - 3*2^(1/3) - 1)/(2^(5/3) - 2^(1/3) + 1)) * (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = 1.42776088919948241359... .
%e A379581 Fractions begin with 1, 1/2, 5/6, -1/6, 1/30, -2/15, 1/105, -104/105, 1/105, -19/210, 1/2310, -769/2310, ...
%t A379581 f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/powfree[n], {n, 1, 50}]]]
%o A379581 (PARI) powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
%o A379581 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powfree(k); print1(numerator(s), ", "))};
%Y A379581 Cf. A055231, A328013, A370900, A370901, A379579, A379582 (denominators).
%K A379581 sign,easy,frac
%O A379581 1,3
%A A379581 _Amiram Eldar_, Dec 26 2024