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A379365 Numerators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).

%I A379365 #8 Dec 22 2024 16:53:39
%S A379365 1,2,13,89,307,283,4039,761,5639,16189,17125,10396,54437,52862,54227,
%T A379365 847157,9646327,9474727,361375699,355820149,27844153,27355753,
%U A379365 28039513,27731821,366667513,72266837,219763471,217455781,4211659759,835576403,51882159671,25692722941
%N A379365 Numerators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).
%H A379365 Amiram Eldar, <a href="/A379365/b379365.txt">Table of n, a(n) for n = 1..1000</a>
%H A379365 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 4.5, pp. 23-24.
%F A379365 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A018804(k)).
%F A379365 a(n)/A379366(n) = Sum_{j=0..N} D_j/log(n)^(j-1/2) + O(1/log(n)^(N+1/2), for any integer N >= 1, where D_j are constants, and in particular D_0 = (1/(4*log(2)-2)-1) * (2/sqrt(Pi)) * Product_{p prime} (sqrt(1-1/p) * Sum_{k>=1} 1/A018804(p^k)) = 0.38291621042855537524... .
%e A379365 Fractions begin with 1, 2/3, 13/15, 89/120, 307/360, 283/360, 4039/4680, 761/936, 5639/6552, 16189/19656, 17125/19656, 10396/12285, ...
%t A379365 f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/pillai[n], {n, 1, 50}]]]
%o A379365 (PARI) pillai(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 A379365 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / pillai(k); print1(numerator(s), ", "))};
%Y A379365 Cf. A018804, A272718, A370895, A379363, A379366 (denominators).
%K A379365 nonn,easy,frac
%O A379365 1,2
%A A379365 _Amiram Eldar_, Dec 21 2024