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Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.
(Formerly M4185 N1743)
114

%I M4185 N1743 #170 Sep 07 2024 15:39:03

%S 1,6,28,140,270,496,672,1638,2970,6200,8128,8190,18600,18620,27846,

%T 30240,32760,55860,105664,117800,167400,173600,237510,242060,332640,

%U 360360,539400,695520,726180,753480,950976,1089270,1421280,1539720

%N Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

%C Note that the harmonic mean of the divisors of k = k*tau(k)/sigma(k).

%C 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).

%C Equivalently, the average of the divisors of k divides k.

%C 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

%C 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.

%C 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

%C 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

%C Conjecture: All terms > 1 are Zumkeller numbers (A083207). Verified for all n <= 50. - _Ivan N. Ianakiev_, Nov 22 2017

%C Verified for n <= 937. - _David A. Corneth_, Jun 07 2020

%C Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0. - _Amiram Eldar_, Jun 01 2020

%C 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

%C Conjecture: all terms > 1 have a Mersenne prime as a factor. - _Ivan Borysiuk_, Jan 28 2024

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

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.

%D W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Robert G. Wilson v, <a href="/A001599/b001599.txt">Table of n, a(n) for n = 1..937</a> (terms n = 1..170 from T. D. Noe and Klaus Brockhaus)

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, <a href="http://arxiv.org/abs/1601.03081">The Biharmonic mean</a>, arXiv:1601.03081 [math.NT], 2016.

%H Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.

%H Ivan Borysiuk, <a href="/A001599/a001599_1.txt">Conjectured to be the 10000 smallest Ore numbers</a>

%H Graeme L. Cohen, <a href="/A007340/a007340.pdf">Email to N. J. A. Sloane, Apr. 1994</a>.

%H Graeme L. Cohen, <a href="https://doi.org/10.1090/S0025-5718-97-00819-3">Numbers whose positive divisors have small integral harmonic mean</a>, Mathematics of Computation, Vol. 66, No. 218, (1997), pp. 883-891.

%H Graeme L. Cohen and Ronald M. Sorli, <a href="http://www.fq.math.ca/Scanned/36-5/cohen.pdf">Harmonic seeds</a>, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4.

%H Graeme L. Cohen and Ronald M. Sorli, <a href="http://dx.doi.org/10.1090/S0025-5718-10-02337-9">Odd harmonic numbers exceed 10^24</a>, Math. Comp., Vol. 79, No. 272 (2010), pp. 2451-2460.

%H Mariano Garcia, <a href="http://www.jstor.org/stable/2307792">On numbers with integral harmonic mean</a>, Amer. Math. Monthly, Vol. 61, No. 2 (1954), pp. 89-96.

%H Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/html/harmonic-e.html#mark1">All harmonic numbers less than 10^14</a>.

%H Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/files/list4">Table of a(n) for n = 1..937</a>.

%H T. Goto and S. Shibata, <a href="http://dx.doi.org/10.1090/S0025-5718-03-01554-0">All numbers whose positive divisors have integral harmonic mean up to 300</a>, Math. Comput., Vol. 73, No. 245 (2004), pp. 475-491.

%H Richard K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>.

%H Hans-Joachim Kanold, <a href="http://dx.doi.org/10.1007/BF01342887">Über das harmonische Mittel der Teiler einer natürlichen Zahl</a>, Math. Ann., Vol. 133 (1957), pp. 371-374.

%H Oystein Ore, <a href="http://www.jstor.org/stable/2305616">On the averages of the divisors of a number</a>, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.

%H Oystein Ore, <a href="/A001599/a001599.pdf">On the averages of the divisors of a number</a>. (annotated scanned copy)

%H Carl Pomerance, On a Problem of Ore: Harmonic Numbers, unpublished manuscript, 1973; abstract *709-A5, Notices of the American Mathematical Society, Vol. 20, 1973, page A-648, <a href="https://www.ams.org/journals/notices/197311/197311FullIssue.pdf">entire volume</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicDivisorNumber.html">Harmonic Divisor Number</a>.

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Harmonic_mean">Harmonic mean</a>.

%H Wikipedia, <a href="http://www.wikipedia.org/wiki/Harmonic_divisor_number">Harmonic divisor number</a>.

%H Andreas and Eleni Zachariou, <a href="http://www.hms.gr/apothema/?s=sa&amp;i=261">Perfect, semi-perfect and Ore numbers</a>, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; <a href="https://eudml.org/doc/238923">alternative link</a>.

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%F { k : A106315(k) = 0 }. - _R. J. Mathar_, Jan 25 2017

%e 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.

%e 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.

%p q:= (p,k) -> p^k*(p-1)*(k+1)/(p^(k+1)-1):

%p filter:= proc(n) local t; mul(q(op(t)),t=ifactors(n)[2])::integer end proc:

%p select(filter, [$1..10^6]); # _Robert Israel_, Jan 14 2016

%t Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]

%t Select[Range[1600000],IntegerQ[HarmonicMean[Divisors[#]]]&] (* _Harvey P. Dale_, Oct 20 2012 *)

%o (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 */

%o (Haskell)

%o import Data.Ratio (denominator)

%o import Data.List (genericLength)

%o a001599 n = a001599_list !! (n-1)

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

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

%o where ds = a027750_row x

%o -- _Reinhard Zumkeller_, Jun 04 2013, Jan 20 2012

%o (GAP) Concatenation([1],Filtered([2,4..2000000],n->IsInt(n*Tau(n)/Sigma(n)))); # _Muniru A Asiru_, Nov 26 2018

%o (Python)

%o from sympy import divisor_sigma as sigma

%o def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0

%o print([n for n in range(1, 10**4) if ok(n)]) # _Michael S. Branicky_, Jan 06 2021

%o (Python)

%o from itertools import count, islice

%o from functools import reduce

%o from math import prod

%o from sympy import factorint

%o def A001599_gen(startvalue=1): # generator of terms >= startvalue

%o for n in count(max(startvalue,1)):

%o f = factorint(n)

%o s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())

%o if not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:

%o yield n

%o A001599_list = list(islice(A001599_gen(),20)) # _Chai Wah Wu_, Feb 14 2023

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

%Y See A001600 and A090240 for the integer values obtained.

%Y sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).

%Y sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).

%Y Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.

%Y Cf. A027750.

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Klaus Brockhaus_, Sep 18 2001