This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A057020 #54 Jul 18 2025 14:46:29
%S A057020 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 A057020 15,9,16,21,12,27,12,91,19,15,14,45,21,12,22,14,13,18,24,62,19,31,18,
%U A057020 49,27,15,18,15,20,45,30,14,31,24,52,127,21,18,34,21,24,18
%N A057020 Numerator of (sum of divisors of n / number of divisors of n).
%C A057020 Numerator of arithmetic mean of the divisors of n. - _Jaroslav Krizek_, Apr 26 2010
%C A057020 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
%D A057020 V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.
%H A057020 Reinhard Zumkeller, <a href="/A057020/b057020.txt">Table of n, a(n) for n = 1..10000</a>
%H A057020 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 A057020 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 A057020 Marcin Mazur and Bogdan V. Petrenko, <a href="https://doi.org/10.1007/s11139-013-9546-3">Representations of analytic functions as infinite products and their application to numerical computations</a>, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; <a href="https://arxiv.org/abs/1202.1335">arXiv preprint</a>, arXiv:1202.1335 [math.NT], 2012.
%H A057020 Daniel Sutantyo, <a href="https://web.archive.org/web/20130730210221/https://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. [Wayback Machine link]
%F A057020 a(n) = A057021(n) * A000203(n)/A000005(n) = A000203(n)/A009205(n) = (A057022(n) + A054025(n)/A000005(n)) * A057021(n).
%F A057020 Sum_{k=1..n} a(k)/A057021(k) ~ c * n^2 /sqrt(log(n)), where c = A308051. - _Amiram Eldar_, Apr 15 2025
%e A057020 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 A057020 with(numtheory): seq(numer(sigma(n)/tau(n)), n=1..70) ; # _Zerinvary Lajos_, Jun 04 2008
%t A057020 Numerator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* _Alonso del Arte_, Feb 24 2006 *)
%t A057020 Table[Numerator[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,100}] (* _Harvey P. Dale_, Dec 19 2023 *)
%o A057020 (Haskell)
%o A057020 import Data.Ratio ((%), numerator)
%o A057020 a057020 n = numerator $ a000203 n % a000005 n
%o A057020 -- _Reinhard Zumkeller_, Jan 06 2012
%o A057020 (PARI) a(n)=numerator(sigma(n)/numdiv(n)) \\ _Charles R Greathouse IV_, May 17 2012
%o A057020 (SageMath) [numerator(sigma(n, 1)/sigma(n, 0)) for n in range(1, 71)] # _Stefano Spezia_, Jul 18 2025
%Y A057020 Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022, A308051.
%K A057020 frac,nonn
%O A057020 1,2
%A A057020 _Henry Bottomley_, Jul 21 2000