A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

1, 2, 11, 43, 53, 4, 37, 293, 329, 103, 113, 107, 809, 129, 809, 12913, 41119, 39691, 41833, 8081, 33395, 32443, 33871, 10973, 148361, 48275, 7149, 34861, 108119, 319937, 164941, 1761311, 112361, 662011, 5405483, 26502319, 516671461, 508357441, 3620857237, 3556192637

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

a(n)/A379516(n) = E * log(n) + F + O(log(n)^(5/3) * log(log(n))^(4/3) / n^u), where u > 0, E = A308041 * (2/(A323482 + 1/2) - 1) = 0.10259754363391420806..., and F is a constant.

Example

Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 293/360, 329/360, 103/120, 113/120, 107/120, ...

Mathematica

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

Prog

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

Crossrefs

Cf. A034448, A064609, A308041, A323482, A370898, A379513, A379516 (denominators).

Sequence in context: A140322 A027247 A379617 * A141190 A048500 A387868

Adjacent sequences: A379512 A379513 A379514 * A379516 A379517 A379518

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 23 2024

Status

approved