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%I #44 Dec 19 2023 17:32:24
%S 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,
%T 15,9,16,21,12,27,12,91,19,15,14,45,21,12,22,14,13,18,24,62,19,31,18,
%U 49,27,15,18,15,20,45,30,14,31,24,52,127,21,18,34,21,24,18
%N Numerator of (sum of divisors of n / number of divisors of n).
%C Numerator of arithmetic mean of the divisors of n. - _Jaroslav Krizek_, Apr 26 2010
%C 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
%H Reinhard Zumkeller, <a href="/A057020/b057020.txt">Table of n, a(n) for n = 1..10000</a>
%H V. Arnold, <a href="http://dx.doi.org/10.1007/s00021-004-0130-x">Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics</a>, Journal of Mathem. Fluid Mechanics 7 (2005), pp. S4-S50.
%H Paul T. Bateman, Paul Erdős, Carl Pomerance, and Ernst G. Straus, <a href="http://www.renyi.hu/~p_erdos/1981-37.pdf">The arithmetic mean of the divisors of an integer</a> in Analytic Number Theory (1980), pp. 197-220.
%H Daniel Sutantyo, <a href="http://web.science.mq.edu.au/groups/acac/researchstudent_completed/Daniel_Sutantyo.pdf">Elementary and Analytic Methods in Number Theory</a>, M.S. thesis (Macquarie University, 2007), chapter 3.
%F a(n) = A057021(n) * A000203(n)/A000005(n) = A000203(n)/A009205(n) = (A057022(n) + A054025(n)/A000005(n)) * A057021(n).
%e 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.
%p with(numtheory): seq(numer(sigma(n)/tau(n)), n=1..70) ; # _Zerinvary Lajos_, Jun 04 2008
%t Numerator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* _Alonso del Arte_, Feb 24 2006 *)
%t Table[Numerator[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,100}] (* _Harvey P. Dale_, Dec 19 2023 *)
%o (Haskell)
%o import Data.Ratio ((%), numerator)
%o a057020 n = numerator $ a000203 n % a000005 n
%o -- _Reinhard Zumkeller_, Jan 06 2012
%o (PARI) a(n)=numerator(sigma(n)/numdiv(n)) \\ _Charles R Greathouse IV_, May 17 2012
%Y Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022.
%K frac,nonn
%O 1,2
%A _Henry Bottomley_, Jul 21 2000
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