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%I M2389 #104 Oct 09 2023 12:52:00
%S 1,3,5,6,7,11,13,14,15,17,19,20,21,22,23,27,29,30,31,33,35,37,38,39,
%T 41,42,43,44,45,46,47,49,51,53,54,55,56,57,59,60,61,62,65,66,67,68,69,
%U 70,71,73,77,78,79,83,85,86,87,89,91,92,93,94,95,96,97,99,101,102,103,105
%N Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).
%C Sometimes called arithmetic numbers.
%C Generalized (sigma_r)-numbers are numbers j for which sigma_r(j)/sigma_0(j) = c^r. Sigma_r(j) denotes the sum of the r-th powers of the divisors of j; c,r are positive integers. The numbers in this sequence are sigma_1-numbers; those in A140480 are sigma_2-numbers. - _Ctibor O. Zizka_, Jul 14 2008
%C {a(n)} = union A175678 and A175679 where A175678 = numbers m such that the arithmetic mean Ad(m) of divisors of m and the arithmetic mean Ah(m) of numbers h < m such that gcd(h,m) = 1 are both integers and A175679 = numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ak(m) of the numbers k <= m are both integers. - _Jaroslav Krizek_, Aug 07 2010
%C All odd primes (A065091) are arithmetic numbers. - _Wesley Ivan Hurt_, Oct 04 2013
%C A069928(n) = number of arithmetic numbers not greater than n. - _Reinhard Zumkeller_, Jul 28 2014
%C A102187(n) divides a(n) for a(n) = 1, 6, 140, 270, 672, ... A007340. - _Thomas Ordowski_, Oct 24 2014
%C The quotients sigma(j)/tau(j) are in A102187. - _Bernard Schott_, Jun 07 2017
%D R. K. Guy, Unsolved Problems in Number Theory, B2.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003601/b003601.txt">Table of n, a(n) for n = 1..10000</a>
%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="http://arxiv.org/abs/1601.03081">The Biharmonic mean</a>, arXiv:1601.03081 [math.NT], 2016.
%H Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, <a href="https://math.dartmouth.edu/~carlp/PDF/31.pdf">The arithmetic mean of the divisors of an integer</a> (1981). In Knopp, M.I. ed., Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. 899. Springer-Verlag., pp. 197-220.
%H Antonio M. Oller-Marcén, <a href="http://arxiv.org/abs/1206.1823">On arithmetic numbers</a>, arXiv:1206.1823 [math.NT], 2012.
%H O. Ore, <a href="http://www.jstor.org/stable/2305616">On the averages of the divisors of a number</a>, Amer. Math. Monthly, 55 (1948), 615-619.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Arithmetic_number">Arithmetic number</a>.
%F a(n) ~ n. - _Charles R Greathouse IV_, Jul 10 2012
%F A245656(a(n)) = 1. - _Reinhard Zumkeller_, Jul 28 2014
%e Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - _Bernard Schott_, Jun 07 2017
%p with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod tau(n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # corrected by _Wesley Ivan Hurt_, Oct 03 2013
%t Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* _Stefan Steinerberger_, Apr 03 2006 *)
%o (Haskell)
%o a003601 n = a003601_list !! (n-1)
%o a003601_list = filter ((== 1) . a245656) [1..]
%o -- _Reinhard Zumkeller_, Jul 28 2014, Dec 31 2013, Jan 06 2012
%o (PARI) is(n)=sigma(n)%numdiv(n)==0 \\ _Charles R Greathouse IV_, Jul 10 2012
%o (Python)
%o from sympy import divisors, divisor_count
%o [n for n in range(1,10**5) if not sum(divisors(n)) % divisor_count(n)] # _Chai Wah Wu_, Aug 05 2014
%o (GAP) a:=Filtered([1..110],n->Sigma(n) mod Tau(n)=0);; Print(a); # _Muniru A Asiru_, Jan 25 2019
%Y Complement is A049642.
%Y Cf. A000005, A000203, A054025, A001599, A007340, A140480, A102187.
%Y Cf. A245644, A245656, A069928. Nonprimes are in A023883.
%K nonn,nice,easy
%O 1,2
%A _N. J. A. Sloane_, _Mira Bernstein_
%E _David W. Wilson_, Oct 15 1996, points out that 30 was missing.
%E More terms from _Stefan Steinerberger_, Apr 03 2006
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