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A001599
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Harmonic or Ore numbers: numbers n such that harmonic mean of divisors of n is an integer.
(Formerly M4185 N1743)
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32
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1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720
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OFFSET
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1,2
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COMMENTS
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Note that the harmonic mean of the divisors of n = n*tau(n)/sigma(n).
Equivalently, n*tau(n)/sigma(n) is an integer, where tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).
Equivalently, the average of the divisors of n divides n.
Ore showed that every perfect number (A000396) is harmonic. The converse does not hold: 140 is harmonic but not perfect. Ore conjectured that 1 is the only odd harmonic number.
Other example of power mean numbers n such that power mean of divisors of n is an integer are RMS numbers A140480. [From Ctibor O.Zizka (c.zizka(AT)email.cz), Sep 20 2008]
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REFERENCES
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G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 386-390; errata, 39 (2001) 4.
M. Garcia, On numbers with integral harmonic mean. Amer. Math. Monthly 61, (1954). 89-96.
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
R. K. Guy, Unsolved Problems in Number Theory, B2.
H.-J. Kanold, Uber das harmonische Mittel der Teiler einer natuerlichen Zahl, Math. Ann., 133 (1957) 371-374.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract *709-A5, Notices Amer. Math. Soc., 20 (1973) Abstract A-648.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grece (N.S.), 13 (1972) 12-22.
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LINKS
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T. D. Noe and Klaus Brockhaus, Table of n, a(n) for n=1..170
Takeshi Goto, All harmonic numbers less than 10^14
Takeshi Goto, Table of a(n) for n=1..937
Graeme L. Cohen, Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp. 79 (272) (2010) 2451-2460. [From R. J. Mathar, Oct 08 2010]
Eric Weisstein's World of Mathematics, Harmonic Mean
Eric Weisstein's World of Mathematics, Harmonic Divisor Number
Wikipedia, Harmonic mean
Wikipedia, Harmonic divisor number
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EXAMPLE
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n=140 has Sigma[ 0,140 ]=12 divisors with Sigma[ 1,140 ]=336. Average divisor is 336/12=28, an integer and divides n: n=5*28. n=496, Sigma[ 0,496 ]=10, Sigma[ 1,496 ]=992: average divisor 99.2 is not an integer, but n/(Sigma_1/Sigma_0)=496/99.2=5 is an integer.
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MATHEMATICA
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Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
Select[Range[1600000], IntegerQ[HarmonicMean[Divisors[#]]]&] (* Harvey P. Dale, Oct 20 2012 *)
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PROG
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(PARI) a(n)=if(n<0, 0, n=a(n-1); until(0==(sigma(n, 0)*n)%sigma(n, 1), n++); n) - Michael Somos Feb 06 2004
(Haskell)
import Data-Ratio ((%), denominator)
a001599_list = filter ((== 1) . denominator . hm) [1..] where
hm x = fromIntegral (length divs) * recip (sum $ map (1 %) divs) where
divs = [d | d <- [1..x], mod x d == 0]
-- Reinhard Zumkeller, Jan 20 2012
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CROSSREFS
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See A003601 for analogues referring to arithmetic mean and A000290 for geometric mean of divisors.
See A001600 and A090240 for the integer values obtained.
sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).
sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).
Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.
Sequence in context: A208439 A108051 A199315 * A074247 A053783 A216383
Adjacent sequences: A001596 A001597 A001598 * A001600 A001601 A001602
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Klaus Brockhaus, Sep 18 2001
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STATUS
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approved
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