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

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			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, ...
		

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.

Crossrefs

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

Cf. A059956, A073004.

Programs

Mathematica

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

PARI

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

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).