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A057020 Numerator of (sum of divisors of n / number of divisors of n). 14
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
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 *)
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
CROSSREFS
Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022.
Sequence in context: A182891 A071190 A295314 * A257322 A276466 A289336
KEYWORD
frac,nonn
AUTHOR
Henry Bottomley, Jul 21 2000
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)