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A379619
Numerators of the partial sums of the reciprocals of the alternating sum of divisors function (A206369).
3
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
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
KEYWORD
nonn,easy,frac
AUTHOR
Amiram Eldar, Dec 27 2024
STATUS
approved