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

1, 2, 13, 89, 307, 283, 4039, 761, 5639, 16189, 17125, 10396, 54437, 52862, 54227, 847157, 9646327, 9474727, 361375699, 355820149, 27844153, 27355753, 28039513, 27731821, 366667513, 72266837, 219763471, 217455781, 4211659759, 835576403, 51882159671, 25692722941

Offset

1,2

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.

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A018804(k)).

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... .

Example

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, ...

Mathematica

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}]]]

Prog

(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(numerator(s), ", "))};

Crossrefs

Cf. A018804, A272718, A370895, A379363, A379366 (denominators).

Sequence in context: A300429 A106938 A106937 * A033891 A172968 A126035

Adjacent sequences: A379362 A379363 A379364 * A379366 A379367 A379368

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 21 2024

Status

approved