A003601 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)).

1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 105

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• nice
• easy

Comments

Sometimes called arithmetic numbers.

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

{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

All odd primes (A065091) are arithmetic numbers. - Wesley Ivan Hurt, Oct 04 2013

A069928(n) = number of arithmetic numbers not greater than n. - Reinhard Zumkeller, Jul 28 2014

A102187(n) divides a(n) for a(n) = 1, 6, 140, 270, 672, ... A007340. - Thomas Ordowski, Oct 24 2014

The quotients sigma(j)/tau(j) are in A102187. - Bernard Schott, Jun 07 2017

Examples

			Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - _Bernard Schott_, Jun 07 2017
		

References

• R. K. Guy, Unsolved Problems in Number Theory, B2.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
• Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer (1981). In Knopp, M.I. ed., Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. 899. Springer-Verlag., pp. 197-220.
• Antonio M. Oller-Marcén, On arithmetic numbers, arXiv:1206.1823 [math.NT], 2012.
• O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
• Wikipedia, Arithmetic number.

Crossrefs

Complement is A049642.

Cf. A000005, A000203, A054025, A001599, A007340, A140480, A102187.

Cf. A245644, A245656, A069928. Nonprimes are in A023883.

Programs

GAP

a:=Filtered([1..110],n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019

Haskell

a003601 n = a003601_list !! (n-1)
a003601_list = filter ((== 1) . a245656) [1..]
-- Reinhard Zumkeller, Jul 28 2014, Dec 31 2013, Jan 06 2012

Maple

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

Mathematica

Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* Stefan Steinerberger, Apr 03 2006 *)

PARI

is(n)=sigma(n)%numdiv(n)==0 \\ Charles R Greathouse IV, Jul 10 2012

Python

from sympy import divisors, divisor_count
[n for n in range(1,10**5) if not sum(divisors(n)) % divisor_count(n)] # Chai Wah Wu, Aug 05 2014

Formula

a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012

A245656(a(n)) = 1. - Reinhard Zumkeller, Jul 28 2014

Extensions

David W. Wilson, Oct 15 1996, points out that 30 was missing.

More terms from Stefan Steinerberger, Apr 03 2006