

A003601


Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n). Alternatively, tau(n) (A000005(n)) divides sigma(n) (A000203(n)).
(Formerly M2389)


55



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
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OFFSET

1,2


COMMENTS

Sometimes called arithmetic numbers.
Generalized (sigma_r)numbers are numbers n for which sigma_r(n)/sigma_0(n) = c^r. Sigma_r(n) denotes sum of rth powers of divisors of n; c,r positive integers. The numbers in this sequence are sigma_1numbers; those in A140480 are sigma_2numbers.  Ctibor O. Zizka, Jul 14 2008
a(n) = union A175678(n) and A175679(n+1) where A175678 = numbers m such that arithmetic mean Ad(m) of divisors of m and arithmetic mean Ah(m) of numbers h < m such that GCD(h,m) = 1 both integer and A175679 = numbers m such that arithmetic mean Ad(m) of divisors of m and arithmetic mean Ak(m) of numbers k <= m both integer.  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


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, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Antonio M. OllerMarcĂ©n, 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), 615619.
Wikipedia, Arithmetic number


FORMULA

a(n) ~ n.  Charles R Greathouse IV, Jul 10 2012
A245656(a(n)) = 1.  Reinhard Zumkeller, Jul 28 2014


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 *)


PROG

(Haskell)
a003601 n = a003601_list !! (n1)
a003601_list = filter ((== 1) . a245656) [1..]
 Reinhard Zumkeller, Jul 28 2014, Dec 31 2013, Jan 06 2012
(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


CROSSREFS

Complement is A049642.
Cf. A000005, A000203, A054025, A001599, A007340, A140480.
Cf. A245644, A245656, A069928. Nonprimes are in A023883.
Sequence in context: A242076 A064728 A046839 * A216782 A072600 A047582
Adjacent sequences: A003598 A003599 A003600 * A003602 A003603 A003604


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane, Mira Bernstein


EXTENSIONS

David W. Wilson, Oct 15 1996, points out that 30 was missing.
More terms from Stefan Steinerberger, Apr 03 2006


STATUS

approved



