A379366 Denominators of the partial alternating sums of the reciprocals of Pillai's arithmetical function (A018804).
1, 3, 15, 120, 360, 360, 4680, 936, 6552, 19656, 19656, 12285, 61425, 61425, 61425, 982800, 10810800, 10810800, 399999600, 399999600, 30769200, 30769200, 30769200, 30769200, 399999600, 79999920, 239999760, 239999760, 4559995440, 911999088, 55631944368, 27815972184
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
- László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.5, pp. 23-24.
Programs
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Mathematica
f[p_, e_] := (e*(p-1)/p + 1)*p^e; pillai[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/pillai[n], {n, 1, 50}]]] -
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]);} list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / pillai(k); print1(denominator(s), ", "))};
Formula
a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A018804(k)).