cp's OEIS Frontend

A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.

%I A001599 M4185 N1743 #177 Jul 18 2025 11:39:16
%S A001599 1,6,28,140,270,496,672,1638,2970,6200,8128,8190,18600,18620,27846,
%T A001599 30240,32760,55860,105664,117800,167400,173600,237510,242060,332640,
%U A001599 360360,539400,695520,726180,753480,950976,1089270,1421280,1539720
%N A001599 Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.
%C A001599 Note that the harmonic mean of the divisors of k = k*tau(k)/sigma(k).
%C A001599 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 A001599 Equivalently, the average of the divisors of k divides k.
%C A001599 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 A001599 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 A001599 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 A001599 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 A001599 Conjecture: All terms > 1 are Zumkeller numbers (A083207). Verified for all n <= 50. - _Ivan N. Ianakiev_, Nov 22 2017
%C A001599 Verified for n <= 937. - _David A. Corneth_, Jun 07 2020
%C A001599 Kanold (1957) proved that the asymptotic density of the harmonic numbers is 0. - _Amiram Eldar_, Jun 01 2020
%C A001599 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 A001599 Conjecture: all terms > 1 have a Mersenne prime as a factor. - _Ivan Borysiuk_, Jan 28 2024
%D A001599 G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
%D A001599 Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
%D A001599 W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
%D A001599 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001599 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A001599 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.
%H A001599 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 A001599 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 A001599 Abiodun E. Adeyemi, <a href="https://arxiv.org/abs/1906.05798">A Study of @-numbers</a>, arXiv:1906.05798 [math.NT], 2019.
%H A001599 Ivan Borysiuk, <a href="/A001599/a001599_1.txt">Conjectured to be the 10000 smallest Ore numbers</a>
%H A001599 Graeme L. Cohen, <a href="/A007340/a007340.pdf">Email to N. J. A. Sloane, Apr. 1994</a>.
%H A001599 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 A001599 Graeme L. Cohen and Ronald M. Sorli, <a href="https://web.archive.org/web/2024*/https://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 A001599 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 A001599 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 A001599 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 A001599 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 A001599 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 A001599 Richard K. Guy, <a href="/A001599/a001599_1.pdf">Letter to N. J. A. Sloane with attachment, Jun. 1991</a>.
%H A001599 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 A001599 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 A001599 Oystein Ore, <a href="/A001599/a001599.pdf">On the averages of the divisors of a number</a>. (annotated scanned copy)
%H A001599 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 A001599 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicMean.html">Harmonic Mean</a>.
%H A001599 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicDivisorNumber.html">Harmonic Divisor Number</a>.
%H A001599 Wikipedia, <a href="http://www.wikipedia.org/wiki/Harmonic_mean">Harmonic mean</a>.
%H A001599 Wikipedia, <a href="http://www.wikipedia.org/wiki/Harmonic_divisor_number">Harmonic divisor number</a>.
%H A001599 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 A001599 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F A001599 { k : A106315(k) = 0 }. - _R. J. Mathar_, Jan 25 2017
%e A001599 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 A001599 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 A001599 q:= (p,k) -> p^k*(p-1)*(k+1)/(p^(k+1)-1):
%p A001599 filter:= proc(n) local t; mul(q(op(t)),t=ifactors(n)[2])::integer end proc:
%p A001599 select(filter, [$1..10^6]); # _Robert Israel_, Jan 14 2016
%t A001599 Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], {n, 1, 1550000}]
%t A001599 Select[Range[1600000],IntegerQ[HarmonicMean[Divisors[#]]]&] (* _Harvey P. Dale_, Oct 20 2012 *)
%o A001599 (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 A001599 (Haskell)
%o A001599 import Data.Ratio (denominator)
%o A001599 import Data.List (genericLength)
%o A001599 a001599 n = a001599_list !! (n-1)
%o A001599 a001599_list = filter ((== 1) . denominator . hm) [1..] where
%o A001599    hm x = genericLength ds * recip (sum $ map (recip . fromIntegral) ds)
%o A001599           where ds = a027750_row x
%o A001599 -- _Reinhard Zumkeller_, Jun 04 2013, Jan 20 2012
%o A001599 (GAP) Concatenation([1],Filtered([2,4..2000000],n->IsInt(n*Tau(n)/Sigma(n)))); # _Muniru A Asiru_, Nov 26 2018
%o A001599 (Python)
%o A001599 from sympy import divisor_sigma as sigma
%o A001599 def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
%o A001599 print([n for n in range(1, 10**4) if ok(n)]) # _Michael S. Branicky_, Jan 06 2021
%o A001599 (Python)
%o A001599 from itertools import count, islice
%o A001599 from functools import reduce
%o A001599 from math import prod
%o A001599 from sympy import factorint
%o A001599 def A001599_gen(startvalue=1): # generator of terms >= startvalue
%o A001599     for n in count(max(startvalue,1)):
%o A001599         f = factorint(n)
%o A001599         s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
%o A001599         if not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
%o A001599             yield n
%o A001599 A001599_list = list(islice(A001599_gen(),20)) # _Chai Wah Wu_, Feb 14 2023
%Y A001599 See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors.
%Y A001599 See A001600 and A090240 for the integer values obtained.
%Y A001599 sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).
%Y A001599 sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).
%Y A001599 Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset of A003601.
%Y A001599 Cf. A027750.
%K A001599 nonn,nice
%O A001599 1,2
%A A001599 _N. J. A. Sloane_
%E A001599 More terms from _Klaus Brockhaus_, Sep 18 2001