

A001599


Harmonic or Ore numbers: numbers n such that the harmonic mean of the divisors of n is an integer.
(Formerly M4185 N1743)


53



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

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
Conjecture: Except for n = 1, a(n) is a Zumkeller number (A083207). I've verified this for all n in [2,50].  Ivan N. Ianakiev, Nov 22 2017


REFERENCES

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161172.
R. K. Guy, Unsolved Problems in Number Theory, B2.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142146.
Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); see Abstract *709A5, Notices Amer. Math. Soc., 20 (1973) Abstract A648.
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).


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)
Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
G. L. Cohen, Email to N. J. A. Sloane, Apr. 1994
Graeme L. Cohen, Numbers whose positive divisors have small integral harmonic mean, Mathematics of Computation, Vol. 66, No. 218, (1997), pp. 883891.
G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 386390; errata, 39 (2001) 4.
Graeme L. Cohen, Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp. 79 (272) (2010) 24512460.
M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61, (1954). 8996.
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), 475491.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
H.J. Kanold, Uber das harmonische Mittel der Teiler einer natÃ¼rlichen Zahl, Math. Ann., 133 (1957) 371374.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615619.
O. Ore, On the averages of the divisors of a number (annotated scanned copy)
Eric Weisstein's World of Mathematics, Harmonic Mean
Eric Weisstein's World of Mathematics, Harmonic Divisor Number
Wikipedia, Harmonic mean
Wikipedia, Harmonic divisor number
Andreas and Eleni Zachariou, Perfect, semiperfect and Ore numbers, Bull. Soc. Math. Grece (N.S.), 13 (1972) 1222.


FORMULA

{n: A106315(n)=0}.  R. J. Mathar, Jan 25 2017


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 has 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.


MAPLE

q:= (p, k) > p^k*(p1)*(k+1)/(p^(k+1)1):
filter:= proc(n) local t; mul(q(op(t)), t=ifactors(n)[2])::integer end proc:
select(filter, [$1..10^6]); # Robert Israel, Jan 14 2016


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(n1); until(0==(sigma(n, 0)*n)%sigma(n, 1), n++); n) /* Michael Somos, Feb 06 2004 */
(Haskell)
import DataRatio (denominator)
import Data.List (genericLength)
a001599 n = a001599_list !! (n1)
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
(GAP) Concatenation([1], Filtered([2, 4..2000000], n>IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018


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



