A379520 Denominators of the partial alternating sums of the reciprocals of the unitary totient function (A047994).

1, 1, 2, 6, 12, 12, 12, 84, 168, 168, 840, 840, 280, 840, 105, 105, 1680, 1680, 5040, 5040, 5040, 5040, 55440, 55440, 55440, 55440, 720720, 80080, 80080, 80080, 240240, 7447440, 7447440, 3723720, 620620, 3723720, 11171160, 11171160, 1396395, 5585580, 2234232, 2234232

Offset: 1

Author

Amiram Eldar, Dec 24 2024

Keywords

• nonn
• easy
• frac

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, A177754, A370899, A379518, A379519 (numerators).

Programs

Mathematica

uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Denominator[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(denominator(s), ", "))};

Formula

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