A379519 Numerators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

1, 0, 1, 1, 5, -1, 1, -5, 11, -31, -71, -211, -47, -281, -22, -29, -359, -569, -1427, -1847, -1427, -1931, -18721, -22681, -20371, -24991, -297163, -37467, -34607, -44617, -125843, -4141373, -3769001, -2117233, -327013, -2117233, -6041389, -6662009, -774568, -3297757

Offset: 1

Author

Amiram Eldar, Dec 24 2024

Keywords

• sign
• easy
• frac

Examples

			Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -5/84, 11/168, -31/168, -71/840, -211/840, ...
		

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 4.10, pp. 30-31.

Crossrefs

Cf. A047994, A065442, A177754, A327837, A370899, A379517, A379520 (denominators).

Programs

Mathematica

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

PARI

uphi(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) / uphi(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A047994(k)).

a(n)/A379520(n) = T * log(n) + U + O(log(n)^(5/3) / n^u), where u > 0, T = A327837 * (2/(A065442 + 1) - 1), and U is a constant.