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

Author

Amiram Eldar, Oct 16 2022

Keywords

• nonn
• frac

Comments

See A357843 for more details.

Links

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

Crossrefs

Cf. A000005, A307704, A357843 (numerators).

Similar sequences: A104529, A211178, A357821.

Programs

Mathematica

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

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

Formula

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