A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

1, 1, 1, 2, 7, 11, 17, 7, 3, 5, 7, 19, 25, 11, 25, 113, 143, 133, 163, 51, 14, 51, 61, 117, 391, 361, 391, 371, 431, 52, 119, 19, 81, 19, 81, 709, 799, 377, 799, 1553, 1733, 211, 467, 226, 467, 889, 979, 961, 1021, 991, 259, 503, 274, 2147, 2237, 274, 1141, 274

Offset

1,4

Links

Table of n, a(n) for n=1..58.

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

Formula

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

a(n)/A357844(n) ~ n * Sum_{k=1..N} B_k/log(n)^(k-1/2) + O(n/log(n)^(N+1/2)), where B_k are constants, and in particular B_1 = (1/log(2) - 1) * (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Tóth, 2017).

Example

Fractions begin with 1, 1/2, 1, 2/3, 7/6, 11/12, 17/12, 7/6, 3/2, 5/4, 7/4, 19/12, ...

Mathematica

Numerator[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(numerator(s), ", "))};
(Python)
from fractions import Fraction
from sympy import divisor_count
def A357843(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1, n+1)).numerator # Chai Wah Wu, Oct 16 2022

Crossrefs

Cf. A000005, A307704, A357844 (denominators).

Similar sequences: A104528, A211177, A357820.

Sequence in context: A190750 A049635 A079140 * A176897 A362770 A166005

Adjacent sequences: A357840 A357841 A357842 * A357844 A357845 A357846

Keyword

nonn,frac

Author

Amiram Eldar, Oct 16 2022

Status

approved