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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)).
(Formerly M2389)
117
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 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
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
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.
FORMULA
a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012
A245656(a(n)) = 1. - Reinhard Zumkeller, Jul 28 2014
EXAMPLE
Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - Bernard Schott, Jun 07 2017
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 !! (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
(GAP) a:=Filtered([1..110], n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019
CROSSREFS
Complement is A049642.
Cf. A245644, A245656, A069928. Nonprimes are in A023883.
Sequence in context: A349897 A064728 A046839 * A328557 A216782 A328952
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
David W. Wilson, Oct 15 1996, points out that 30 was missing.
More terms from Stefan Steinerberger, Apr 03 2006
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)