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

1,2

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 4.13, p. 34.

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.

Example

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, ...

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}]]]

Prog

(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), ", "))};

Crossrefs

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

Sequence in context: A219100 A140322 A027247 * A379515 A141190 A048500

Adjacent sequences: A379614 A379615 A379616 * A379618 A379619 A379620

Keyword

nonn,easy,frac,new

Author

Amiram Eldar, Dec 27 2024

Status

approved