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A003601
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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)
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41
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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
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1,2
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COMMENTS
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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]
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REFERENCES
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R. K. Guy, Unsolved Problems in Number Theory, B2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
Antonio M. Oller-Marcén, On arithmetic numbers, 2012
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FORMULA
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a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012
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MAPLE
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with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod sigma[ 0 ](n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t;
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MATHEMATICA
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Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] - Stefan Steinerberger, Apr 03 2006
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PROG
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(Haskell)
import Data.List (elemIndices)
a003601 n = a003601_list !! (n-1)
a003601_list = map (+ 1) $ elemIndices 0 a054025_list
-- Reinhard Zumkeller, Jan 06 2012
(PARI) is(n)=sigma(n)%numdiv(n)==0 \\ Charles R Greathouse IV, Jul 10 2012
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CROSSREFS
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Complement is A049642. Cf. A000005, A000203, A054025, A001599, A007340, A140480.
Sequence in context: A092559 A064728 A046839 * A216782 A072600 A047582
Adjacent sequences: A003598 A003599 A003600 * A003602 A003603 A003604
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, Mira Bernstein
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EXTENSIONS
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David W. Wilson, Oct 15 1996, points out that 30 was missing.
More terms from Stefan Steinerberger, Apr 03 2006
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STATUS
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approved
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