A379621 Numerators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

1, 0, 1, 1, 5, -1, 1, -7, 11, -47, -13, -61, -29, -157, -209, -3139, -5123, -1109, -2887, -3547, -2887, -3679, -3319, -4111, -26137, -30757, -5597, -2071, -277, -343, -1627, -12269, -2269, -625, -391, -1261, -3629, -3937, -1853, -4979, -19223, -21533, -20873, -21797

Offset: 1

Author

Amiram Eldar, Dec 27 2024

Keywords

• sign
• easy
• frac

Examples

			Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -7/60, 11/420, -47/210, -13/105, -61/210, ...
		

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 4.14, p. 35.

Crossrefs

Cf. A206369, A370905, A370906, A379619, A379622 (denominators).

Programs

Mathematica

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/beta[n], {n, 1, 50}]]]

PARI

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

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A206369(k)).

a(n)/A379622(n) = A * log(n) + B + O(1/n^u), where u > 0, and A and B are constants.