A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

1, 2, 1, 3, 6, 12, 12, 6, 2, 4, 4, 12, 12, 6, 12, 60, 60, 60, 60, 20, 5, 20, 20, 40, 120, 120, 120, 120, 120, 15, 30, 5, 20, 5, 20, 180, 180, 90, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 45, 90, 45, 360, 360, 45, 180, 45, 90, 180, 180, 45, 90, 630

Offset

1,2

Comments

See A357843 for more details.

Links

Table of n, a(n) for n=1..64.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Formula

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/d(k)), where d(k) = A000005(k).

Mathematica

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]

Prog

(PARI) lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(denominator(s), ", "))};
(Python)
from fractions import Fraction
from sympy import divisor_count
def A357844(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1, n+1)).denominator # Chai Wah Wu, Oct 16 2022

Crossrefs

Cf. A000005, A307704, A357843 (numerators).

Similar sequences: A104529, A211178, A357821.

Sequence in context: A020992 A025110 A075346 * A104529 A095024 A049900

Adjacent sequences: A357841 A357842 A357843 * A357845 A357846 A357847

Keyword

nonn,frac

Author

Amiram Eldar, Oct 16 2022

Status

approved