

A001599


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


114



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 k = k*tau(k)/sigma(k).
Equivalently, k*tau(k)/sigma(k) is an integer, where tau(k) (A000005) is the number of divisors of k and sigma(k) is the sum of the divisors of k (A000203).
Equivalently, the average of the divisors of k divides k.
Note that the average of the divisors of k is not necessarily an integer, so the above wording should be clarified as follows: k 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 k such that some power mean of the divisors of k 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: All terms > 1 are Zumkeller numbers (A083207). Verified for all n <= 50.  Ivan N. Ianakiev, Nov 22 2017
Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0.  Amiram Eldar, Jun 01 2020
Zachariou and Zachariou (1972) called these numbers "Ore numbers", after the Norwegian mathematician Øystein Ore (1899  1968), who was the first to study them. Ore (1948) and Garcia (1954) referred to them as "numbers with integral harmonic mean of divisors". The term "harmonic numbers" was used by Pomerance (1973). They are sometimes called "harmonic divisor numbers", or "Ore's harmonic numbers", to differentiate them from the partial sums of the harmonic series.  Amiram Eldar, Dec 04 2020
Conjecture: all terms > 1 have a Mersenne prime as a factor.  Ivan Borysiuk, Jan 28 2024


REFERENCES

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161172.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 7475.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142146.
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

Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386390; errata, 39 (2001) 4.
Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A648, entire volume.


FORMULA



EXAMPLE

k=140 has sigma_0(140)=12 divisors with sigma_1(140)=336. The average divisor is 336/12=28, an integer, and divides k: k=5*28, so 140 is in the sequence.
k=496 has sigma_0(496)=10, sigma_1(496)=992: the average divisor 99.2 is not an integer, but k/(sigma_1/sigma_0)=496/99.2=5 is an integer, so 496 is in the sequence.


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:


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
(GAP) Concatenation([1], Filtered([2, 4..2000000], n>IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018
(Python)
from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
(Python)
from itertools import count, islice
from functools import reduce
from math import prod
from sympy import factorint
def A001599_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue, 1)):
f = factorint(n)
s = prod((p**(e+1)1)//(p1) for p, e in f.items())
if not reduce(lambda x, y:x*y%s, (e+1 for e in f.values()), 1)*n%s:
yield n


CROSSREFS

See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors.
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).


KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



