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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)
48
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 (list; graph; refs; listen; history; text; internal format)
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 r-th powers of divisors of n; c,r positive integers. This sequence are sigma_1-numbers, A140480 are sigma_2-numbers. - 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. [From Jaroslav Krizek, Aug 07 2010]

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

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

Antonio M. Oller-Marcén, On arithmetic numbers, 2012

O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

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;

MATHEMATICA

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

PROG

(Haskell)

a003601 n = a003601_list !! (n-1)

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.

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

Maple code corrected by Wesley Ivan Hurt, Oct 03 2013

STATUS

approved

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Last modified September 17 13:37 EDT 2014. Contains 246846 sequences.