A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

1, 3, 12, 84, 84, 7, 56, 840, 10920, 32760, 32760, 32760, 32760, 16380, 32760, 1015560, 338520, 338520, 338520, 338520, 1354080, 4062240, 4062240, 4062240, 131040, 131040, 131040, 131040, 131040, 43680, 21840, 65520, 32760, 98280, 196560, 196560, 3734640, 3734640

Offset

1,2

Comments

See A357845 for more details.

Links

Table of n, a(n) for n=1..38.

Olivier Bordellès and Benoit Cloitre, An alternating sum involving the reciprocal of certain multiplicative functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3.

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)/sigma(k)), where sigma(k) = A000203(k).

Mathematica

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

Prog

(PARI) lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / sigma(k); print1(denominator(s), ", "))};
(Python)
from fractions import Fraction
from sympy import divisor_sigma
def A357846(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(k)) for k in range(1, n+1)).denominator # Chai Wah Wu, Oct 16 2022

Crossrefs

Cf. A000203, A068762, A357845 (numerators).

Similar sequence: A104529, A212718, A357821.

Sequence in context: A171186 A229421 A212718 * A051549 A245775 A277782

Adjacent sequences: A357843 A357844 A357845 * A357847 A357848 A357849

Keyword

nonn,frac

Author

Amiram Eldar, Oct 16 2022

Status

approved