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A001599 Harmonic or Ore numbers: numbers n such that harmonic mean of divisors of n is an integer.
(Formerly M4185 N1743)
33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

Note that the average of the divisors of n is not necessarily an integer, so the above wording should be clarified as follows: n divided by the average is an integer. See A007340. - Thomas Ordowski, Oct 26 2014

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 examples of power mean numbers n such that some power mean of the divisors of n is an integer are the RMS numbers A140480. - Ctibor O. Zizka, Sep 20 2008

Conjecture: Every harmonic number is practical (A005153). I've verified this refinement of Ore's conjecture for all terms less than 10^14. - Jaycob Coleman, Oct 12 2013

REFERENCES

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.

R. K. Guy, Unsolved Problems in Number Theory, B2.

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Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract *709-A5, Notices Amer.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.

LINKS

T. D. Noe, Klaus Brockhaus and Robert G. Wilson v, Table of n, a(n) for n = 1..937 (terms n=1..170 from T. D. Noe and Klaus Brockhaus)

G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 386-390; errata, 39 (2001) 4.

Graeme L. Cohen, Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp. 79 (272) (2010) 2451-2460.

M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61, (1954). 89-96.

Takeshi Goto, All harmonic numbers less than 10^14

Takeshi Goto, Table of a(n) for n=1..937

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

H.-J. Kanold, Uber das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., 133 (1957) 371-374.

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

Eric Weisstein's World of Mathematics, Harmonic Mean

Eric Weisstein's World of Mathematics, Harmonic Divisor Number

Wikipedia, Harmonic mean

Wikipedia, Harmonic divisor number

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(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)

import Data.List (genericLength)

a001599 n = a001599_list !! (n-1)

a001599_list = filter ((== 1) . denominator . hm) [1..] where

   hm x = genericLength ds * recip (sum $ map (recip . fromIntegral) ds)

          where ds = a027750_row x

-- Reinhard Zumkeller, Jun 04 2013, Jan 20 2012

CROSSREFS

See A003601 for analogs 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.

Cf. A027750.

Sequence in context: A208439 A108051 A199315 * A074247 A053783 A216383

Adjacent sequences:  A001596 A001597 A001598 * A001600 A001601 A001602

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Klaus Brockhaus, Sep 18 2001

STATUS

approved

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Last modified November 22 20:13 EST 2014. Contains 249827 sequences.