A379367 Numerators of the partial sums of the reciprocals of the squarefree kernel function (A007947).

1, 3, 11, 7, 38, 27, 199, 117, 386, 793, 8933, 1553, 20574, 41863, 127591, 71303, 1227166, 2539417, 48759433, 24864701, 25095646, 50632187, 1174239991, 605711068, 125604071, 252924241, 267797099, 19356010, 564511331, 1891973791, 58959268151, 31867258958, 8730535499

Offset

1,2

References

Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions, North-Holland Publishing Company, Amsterdam, Netherlands, 1980. See pp. 16-17.

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000

N. G. de Bruijn, On the number of integers <= x whose prime factors divide n, Illinois Journal of Mathematics, Vol. 6, No. 1 (1962), pp. 137-141.

Olivier Robert and Gérald Tenenbaum, Sur la répartition du noyau d'un entier, Indagationes Mathematicae, Vol. 24, No. 4 (2013), pp. 802-914.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.6, pp. 24-26.

Formula

a(n) = numerator(Sum_{k=1..n} 1/A007947(k)).

a(n)/A379368(n) = exp((1 + o(1)) * sqrt(8*log(n)/log(log(n)))).

a(n)/A379368(n) ~ (1/2) * exp(gamma) * F(log(n)) * log(log(n)), where F(t) = (6/Pi^2) * Sum_{m>=1} min(1,exp(t)/m)/Product_{primes p|m} (p+1).

Example

Fractions begin with 1, 3/2, 11/6, 7/3, 38/15, 27/10, 199/70, 117/35, 386/105, 793/210, 8933/2310, 1553/385, ...

Mathematica

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; Numerator[Accumulate[Table[1/rad[n], {n, 1, 50}]]]

Prog

(PARI) rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / rad(k); print1(numerator(s), ", "))};

Crossrefs

Cf. A007947, A073355, A370896, A379368 (denominators), A379369.

Cf. A059956, A073004.

Sequence in context: A119324 A322364 A250034 * A378678 A006495 A112286

Adjacent sequences: A379364 A379365 A379366 * A379368 A379369 A379370

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 21 2024

Status

approved