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A057020 Numerator of (sum of divisors of n / number of divisors of n). 6
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

FORMULA

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

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 *)

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

CROSSREFS

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

Sequence in context: A182885 A182891 A071190 * A257322 A276466 A241558

Adjacent sequences:  A057017 A057018 A057019 * A057021 A057022 A057023

KEYWORD

frac,nonn

AUTHOR

Henry Bottomley, Jul 21 2000

STATUS

approved

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Last modified July 23 05:58 EDT 2017. Contains 289686 sequences.