A057020 Numerator of (sum of divisors of n / number of divisors of n).

1, 3, 2, 7, 3, 3, 4, 15, 13, 9, 6, 14, 7, 6, 6, 31, 9, 13, 10, 7, 8, 9, 12, 15, 31, 21, 10, 28, 15, 9, 16, 21, 12, 27, 12, 91, 19, 15, 14, 45, 21, 12, 22, 14, 13, 18, 24, 62, 19, 31, 18, 49, 27, 15, 18, 15, 20, 45, 30, 14, 31, 24, 52, 127, 21, 18, 34, 21, 24, 18

Offset

1,2

Comments

Numerator of arithmetic mean of the divisors of n. - Jaroslav Krizek, Apr 26 2010

The average order of a(n)/A057021(n) is asymptotic to n/sqrt(log(n)); see the Bateman et al. link or the Sutantyo link. - Charles R Greathouse IV, May 17 2012

References

V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

V. Arnold, Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics, Journal of Mathem. Fluid Mechanics 7 (2005), pp. S4-S50.

Paul T. Bateman, Paul Erdős, Carl Pomerance, and Ernst G. Straus, The arithmetic mean of the divisors of an integer in Analytic Number Theory (1980), pp. 197-220.

Marcin Mazur and Bogdan V. Petrenko, Representations of analytic functions as infinite products and their application to numerical computations, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; arXiv preprint, arXiv:1202.1335 [math.NT], 2012.

Daniel Sutantyo, Elementary and Analytic Methods in Number Theory, M.S. thesis (Macquarie University, 2007), chapter 3. [Wayback Machine link]

Formula

a(n) = A057021(n) * A000203(n)/A000005(n) = A000203(n)/A009205(n) = (A057022(n) + A054025(n)/A000005(n)) * A057021(n).

Sum_{k=1..n} a(k)/A057021(k) ~ c * n^2 /sqrt(log(n)), where c = A308051. - Amiram Eldar, Apr 15 2025

Example

a(12) = 14 since the 6 factors of 12 are 1, 2, 3, 4, 6 and 12 and 1 + 2 + 3 + 4 + 6 + 12 = 28 and 28/6 = 14/3.

Maple

with(numtheory): seq(numer(sigma(n)/tau(n)), n=1..70) ; # Zerinvary Lajos, Jun 04 2008

Mathematica

Numerator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* Alonso del Arte, Feb 24 2006 *)
Table[Numerator[DivisorSigma[1, n]/DivisorSigma[0, n]], {n, 100}] (* Harvey P. Dale, Dec 19 2023 *)

Prog

(Haskell)
import Data.Ratio ((%), numerator)
a057020 n = numerator $ a000203 n % a000005 n
-- Reinhard Zumkeller, Jan 06 2012
(PARI) a(n)=numerator(sigma(n)/numdiv(n)) \\ Charles R Greathouse IV, May 17 2012
(SageMath) [numerator(sigma(n, 1)/sigma(n, 0)) for n in range(1, 71)] # Stefano Spezia, Jul 18 2025

Crossrefs

Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022, A308051.

Sequence in context: A182891 A071190 A295314 * A257322 A276466 A289336

Adjacent sequences: A057017 A057018 A057019 * A057021 A057022 A057023

Keyword

frac,nonn,changed

Author

Henry Bottomley, Jul 21 2000

Status

approved