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A379714 Partial alternating sums of the number of exponential divisors function (A049419).

%I A379714 #9 Dec 31 2024 20:18:58
%S A379714 1,0,1,-1,0,-1,0,-2,0,-1,0,-2,-1,-2,-1,-4,-3,-5,-4,-6,-5,-6,-5,-7,-5,
%T A379714 -6,-4,-6,-5,-6,-5,-7,-6,-7,-6,-10,-9,-10,-9,-11,-10,-11,-10,-12,-10,
%U A379714 -11,-10,-13,-11,-13,-12,-14,-13,-15,-14,-16,-15,-16,-15,-17,-16
%N A379714 Partial alternating sums of the number of exponential divisors function (A049419).
%H A379714 Amiram Eldar, <a href="/A379714/b379714.txt">Table of n, a(n) for n = 1..10000</a>
%H A379714 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 4.15, p. 36.
%F A379714 a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).
%F A379714 Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .
%t A379714 f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]
%o A379714 (PARI) ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
%o A379714 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};
%Y A379714 Cf. A049419, A065442, A145353, A327837.
%Y A379714 Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.
%K A379714 sign,easy
%O A379714 1,8
%A A379714 _Amiram Eldar_, Dec 30 2024