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

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			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, ...
		

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.

Crossrefs

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

Programs

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}]]]

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), ", "))};

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... .