A379619 Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).

1, 2, 5, 17, 37, 43, 15, 79, 573, 152, 311, 484, 657, 2041, 4187, 46897, 94949, 97589, 295847, 300467, 305087, 310631, 313151, 63739, 9181, 9313, 46961, 47401, 333787, 340717, 68513, 9863, 49711, 25103, 6317, 44549, 89483, 90253, 181661, 183047, 9187, 18605, 18671

Offset

1,2

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.

Formula

a(n) = numerator(Sum_{k=1..n} 1/A206369(k)).

a(n)/A379620(n) = A * log(n) + B + O(n^(-1+eps)) for any eps > 0, where A and B are constants, A = Product_{p prime} ((1-1/p) * (1 + Sum_{k>=1} 1/beta(p^k))) = 1.72360989673744398907... .

Example

Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 79/20, 573/140, 152/35, 311/70, 484/105, ...

Mathematica

f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[1/beta[n], {n, 1, 50}]]]

Prog

(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(numerator(s), ", "))};

Crossrefs

Cf. A206369, A370905, A370906, A379620 (denominators), A379621.

Sequence in context: A028916 A100272 A107630 * A379517 A078523 A078324

Adjacent sequences: A379616 A379617 A379618 * A379620 A379621 A379622

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 27 2024

Status

approved