%I #10 Mar 05 2024 11:52:15

%S 1,-2,3,-5,4,-11,2,-18,3,-24,-3,-43,-18,-57,-12,-60,-27,-90,-53,-125,

%T -60,-123,-78,-178,-113,-188,-107,-211,-154,-289,-228,-340,-235,-334,

%U -217,-385,-312,-423,-298,-478,-397,-592,-507,-675,-486,-621,-528,-768,-635,-830

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

%H Amiram Eldar, <a href="/A370895/b370895.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = Sum_{k=1..n} (-1)^(k+1) * A018804(k).

%F 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 f[p_, e_] := (e*(p - 1)/p + 1)*p^e; pil[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Array[(-1)^(#+1) * pil[#] &, 100]]

%o (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 lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * pil(k); print1(s, ", "))};

%Y Cf. A018804, A272718.

%Y Cf. A001620, A306016.

%Y Similar sequences: A068762, A068773, A307704, A357817, A362028.

%K sign,easy

%O 1,2

%A _Amiram Eldar_, Mar 05 2024