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A370895 Partial alternating sums of Pillai's arithmetical function (A018804).

%I A370895 #10 Mar 05 2024 11:52:15
%S A370895 1,-2,3,-5,4,-11,2,-18,3,-24,-3,-43,-18,-57,-12,-60,-27,-90,-53,-125,
%T A370895 -60,-123,-78,-178,-113,-188,-107,-211,-154,-289,-228,-340,-235,-334,
%U A370895 -217,-385,-312,-423,-298,-478,-397,-592,-507,-675,-486,-621,-528,-768,-635,-830
%N A370895 Partial alternating sums of Pillai's arithmetical function (A018804).
%H A370895 Amiram Eldar, <a href="/A370895/b370895.txt">Table of n, a(n) for n = 1..10000</a>
%H A370895 László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
%F A370895 a(n) = Sum_{k=1..n} (-1)^(k+1) * A018804(k).
%F A370895 a(n) = -(1/Pi^2) * n^2 * (log(n) + 2*gamma - 1/2 - zeta'(2)/zeta(2) - 10*log(2)/3) + O(n^(547/416 + eps)), where gamma is Euler's constant (A001620) (Tóth, 2017).
%t A370895 f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(#+1) * pil[#] &, 100]]
%o A370895 (PARI) pil(n) = {my(f=factor(n)); prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2]);}
%o A370895 lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pil(k); print1(s, ", "))};
%Y A370895 Cf. A018804, A272718.
%Y A370895 Cf. A001620, A306016.
%Y A370895 Similar sequences: A068762, A068773, A307704, A357817, A362028.
%K A370895 sign,easy
%O A370895 1,2
%A A370895 _Amiram Eldar_, Mar 05 2024