A379620 Denominators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).
1, 1, 2, 6, 12, 12, 4, 20, 140, 35, 70, 105, 140, 420, 840, 9240, 18480, 18480, 55440, 55440, 55440, 55440, 55440, 11088, 1584, 1584, 7920, 7920, 55440, 55440, 11088, 1584, 7920, 3960, 990, 6930, 13860, 13860, 27720, 27720, 1386, 2772, 2772, 13860, 3465, 6930, 79695
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
- László Tóth, A survey of the alternating sum-of-divisors function, Acta Universitatis Sapientiae, Mathematica, Vol. 5, No. 1 (2013), pp. 93-107. See p. 101, eq. (17).
- 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.
Programs
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Mathematica
f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[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 / beta(k); print1(denominator(s), ", "))};
Formula
a(n) = denominator(Sum_{k=1..n} 1/A206369(k)).