A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

1, 2, 11, 43, 53, 4, 37, 103, 23, 65, 71, 337, 2539, 1217, 2539, 7337, 7757, 1501, 7883, 7631, 31469, 30629, 31889, 6277, 84625, 82753, 423593, 82753, 426869, 421409, 216847, 213727, 108911, 11899, 24253, 119081, 2317139, 760853, 773203, 6889667, 7037867, 13946059

Offset: 1

Author

Amiram Eldar, Dec 27 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 103/120, 23/24, 65/72, 71/72, 337/360, ...
		

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 4.13, p. 34.

Crossrefs

Cf. A188999, A307159, A370904, A379615, A379618 (denominators).

Programs

Mathematica

f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]

PARI

bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).

a(n)/A379618(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) * n^c), where c = log(9/10)/log(2) = -0.152003..., and A and B are constants.