A001600 -id:A001600 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane

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
Verified for n <= 937. - David A. Corneth, Jun 07 2020
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

			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.

G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder CO, 1972, 142-146.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

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, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Abiodun E. Adeyemi, A Study of @-numbers, arXiv:1906.05798 [math.NT], 2019.
Ivan Borysiuk, Conjectured to be the 10000 smallest Ore numbers
Graeme 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. 883-891.
Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4.
Graeme L. Cohen and Ronald M. Sorli, Odd harmonic numbers exceed 10^24, Math. Comp., Vol. 79, No. 272 (2010), pp. 2451-2460.
Mariano Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly, Vol. 61, No. 2 (1954), pp. 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., Vol. 73, No. 245 (2004), pp. 475-491.
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
Hans-Joachim Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, Vol. 55, No. 10 (1948), pp. 615-619.
Oystein Ore, On the averages of the divisors of a number. (annotated scanned copy)
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, entire volume.
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, semi-perfect and Ore numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.
Index entries for sequences where any odd perfect numbers must occur

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.

Concatenation([1],Filtered([2,4..2000000],n->IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018

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

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

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 *)

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 */

from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
print([n for n in range(1, 10**4) if ok(n)]) # Michael S. Branicky, Jan 06 2021

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)//(p-1) 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
A001599_list = list(islice(A001599_gen(),20)) # Chai Wah Wu, Feb 14 2023

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

More terms from Klaus Brockhaus, Sep 18 2001

A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

Original entry on oeis.org

1, 3, 7, 11, 21, 21, 43, 43, 61, 63, 111, 77, 157, 129, 147, 171, 273, 183, 343, 231, 301, 333, 507, 301, 521, 471, 547, 473, 813, 441, 931, 683, 777, 819, 903, 671, 1333, 1029, 1099, 903, 1641, 903, 1807, 1221, 1281, 1521, 2163, 1197, 2101, 1563, 1911, 1727

Offset: 1

Henry Gould, Oct 15 2000

Also sum of the orders of the elements in a cyclic group with n elements, i.e., row sums of A054531. - Avi Peretz (njk(AT)netvision.net.il), Mar 31 2001
Also inverse Moebius transform of EulerPhi(n^2), A002618.
Sequence is multiplicative with a(p^e) = (p^(2*e+1)+1)/(p+1). Example: a(10) = a(2)*a(5) = 3*21 = 63.
a(n) is the number of pairs (a, b) such that the equation ax = b is solvable in the ring (Zn, +, x). See the Mathematical Reflections link. - Michel Marcus, Jan 07 2017
From Jake Duzyk, Jun 06 2023: (Start)
These are the "contraharmonic means" of the improper divisors of square integers (inclusive of 1 and the square integer itself).
Permitting "Contraharmonic Divisor Numbers" to be defined analogously to Øystein Ore's Harmonic Divisor Numbers, the only numbers for which there exists an integer contraharmonic mean of the divisors are the square numbers, and a(n) is the n-th integer contraharmonic mean, expressible also as the sum of squares of divisors of n^2 divided by the sum of divisors of n^2. That is, a(n) = sigma_2(n^2)/sigma(n^2).
(a(n) = A001157(k)/A000203(k) where k is the n-th number such that A001157(k)/A000203(k) is an integer, i.e., k = n^2.)
This sequence is an analog of A001600 (Harmonic means of divisors of harmonic numbers) and A102187 (Arithmetic means of divisors of arithmetic numbers). (End)

David M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, p. 152.
H. W. Gould and Temba Shonhiwa, Functions of GCD's and LCM's, Indian J. Math. (Allahabad), Vol. 39, No. 1 (1997), pp. 11-35.
H. W. Gould and Temba Shonhiwa, A generalization of Cesaro's function and other results, Indian J. Math. (Allahabad), Vol. 39, No. 2 (1997), pp. 183-194.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
Miriam Mahannah El-Farrah, Expectation Numbers of Cyclic Groups, MS Thesis, Western Kentucky University, August 2015.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 154-5.
Walther Janous, Problem 10829, Amer. Math. Monthly, 107 (2000), p. 753.
Yadollah Marefat, Ali Iranmanesh, and Abolfazl Tehranian, On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.
Mathematical Reflections, Solution to Problem U338, Issue 4, 2015, p 17.
Joachim von zur Gathen, Arnold Knopfmacher, Florian Luca, Lutz G. Lucht, and Igor E. Shparlinski, Average order of cyclic groups, J. Théorie Nombres Bordeaux 16 (1) (2004) 107-123.

Cf. A000010, A000203, A000290, A001157, A018804, A050873, A051193, A054522, A057661, A061255, A065764, A078747, A174405 (partial sums), A176797, A226512.

Cf. A308471.

a057660 n = sum $ map (div n) $ a050873_row n
-- Reinhard Zumkeller, Nov 25 2013

Table[ DivisorSigma[ 2, n^2 ] / DivisorSigma[ 1, n^2 ], {n, 1, 128} ]
Table[Total[Denominator[Range[n]/n]], {n, 55}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (p^(2*e + 1) + 1)/(p + 1); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 21 2020 *)

a(n)=if(n<1,0,sumdiv(n,d,d*eulerphi(d)))

a(n)=sumdivmult(n,d, eulerphi(d)*d) \\ Charles R Greathouse IV, Sep 09 2014

from math import gcd
def A057660(n): return sum(n//gcd(n,k) for k in range(1,n+1)) # Chai Wah Wu, Aug 24 2023

from math import prod
from sympy import factorint
def A057660(n): return prod((p**((e<<1)+1)+1)//(p+1) for p,e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024

a(n) = Sum_{d|n} d*A000010(d) = Sum_{d|n} d*A054522(n,d), sum of d times phi(d) for all divisors d of n, where phi is Euler's phi function.
a(n) = sigma_2(n^2)/sigma_1(n^2) = A001157(A000290(n))/A000203(A000290(n)) = A001157(A000290(n))/A065764(n). - Labos Elemer, Nov 21 2001
a(n) = Sum_{d|n} A000010(d^2). - Enrique Pérez Herrero, Jul 12 2010
a(n) <= (n-1)*n + 1, with equality if and only if n is noncomposite. - Daniel Forgues, Apr 30 2013
G.f.: Sum_{n >= 1} n*phi(n)*x^n/(1 - x^n) = x + 3*x^2 + 7*x^3 + 11*x^4 + .... Dirichlet g.f.: sum {n >= 1} a(n)/n^s = zeta(s)*zeta(s-2)/zeta(s-1) for Re s > 3.  Cf. A078747 and A176797. - Peter Bala, Dec 30 2013
a(n) = Sum_{i=1..n} numerator(n/i). - Wesley Ivan Hurt, Feb 26 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = Sum_{k=1..n} lcm(n,k)/k.
a(n) = Sum_{k=1..n} gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
From Vaclav Kotesovec, Jun 13 2021: (Start)
Sum_{k=1..n} a(k)/k ~ 3*zeta(3)*n^2/Pi^2.
Sum_{k=1..n} k^2/a(k) ~ A345294 * n.
Sum_{k=1..n} k*A000010(k)/a(k) ~ A345295 * n. (End)
Sum_{k=1..n} a(k) ~ 2*zeta(3)*n^3/Pi^2. - Vaclav Kotesovec, Jun 10 2023

More terms from James Sellers, Oct 16 2000

A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.

Original entry on oeis.org

1, 6, 140, 270, 672, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000

Offset: 1

N. J. A. Sloane

Intersection of A001599 and A003601.
The following are also in A046985: 1, 6, 672, 30240, 32760. Also contains multiply perfect (A007691) numbers. - Labos Elemer
The numbers whose average divisor is also a divisor. Ore's harmonic numbers A001599 without the numbers A046999. - Thomas Ordowski, Oct 26 2014, Apr 17 2022
Harmonic numbers k whose harmonic mean of divisors (A001600) is also a divisor of k. - Amiram Eldar, Apr 19 2022

			x = 270: Sigma(0, 270) = 16, Sigma(1, 270) = 720; average divisor a = 720/16 = 45 and integer 45 divides x, x/a = 270/45 = 6, but 270 is not in A007691.

G. L. Cohen, personal communication.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, Curious and interesting numbers, Penguin Books, p. 124.

Donovan Johnson, Table of n, a(n) for n = 1..847
G. L. Cohen, Email to N. J. A. Sloane, Apr. 1994
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
Hisanori Mishima, Factorizations of many number sequences
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

Intersection of A003601 and A001599.
Different from A090945.
Cf. A001600, A007691, A046985-A046987, A046999, A054025.

a007340 n = a007340_list !! (n-1)
a007340_list = filter ((== 0) . a054025) a001599_list
-- Reinhard Zumkeller, Dec 31 2013

filter:= proc(n)
uses numtheory;
local a;
a:= sigma(n)/sigma[0](n);
type(a,integer) and type(n/a,integer);
end proc:
select(filter, [$1..2500000]); # Robert Israel, Oct 26 2014

Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}] (* Labos Elemer *)
ahmQ[n_] := Module[{dn = Divisors[n]}, IntegerQ[Mean[dn]] && IntegerQ[HarmonicMean[dn]]]; Select[Range[2500000], ahmQ] (* Harvey P. Dale, Nov 16 2011 *)

is(n)=my(d=divisors(n),s=vecsum(d)); s%#d==0 && #d*n%s==0 \\ Charles R Greathouse IV, Feb 07 2017

a = Sigma(1, x)/Sigma(0, x) integer and b = x/a also.

More terms from Robert G. Wilson v, Oct 03 2002

Edited by N. J. A. Sloane, Oct 05 2008 at the suggestion of R. J. Mathar

A349473 Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 2, 7, 2, 2, 13, 2, 4, 2, 1, 1, 5, 2, 1, 1, 3, 1, 1, 6, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 8, 2, 1, 3, 3, 1, 1, 9, 2, 1, 6, 2, 1, 1, 1, 2, 2, 2, 4, 1, 1, 11, 3, 5, 2, 2, 2, 1, 1, 2, 2, 2, 10, 2, 1, 2, 3, 3, 1, 1, 14

Offset: 1

Amiram Eldar, Nov 19 2021

For an odd prime p > 3, the p-th row has a length 3 with a(p, 1) = a(p, 2) = 1 and a(p, 3) = (p-1)/2.

For a harmonic number m = A001599(k), the m-th row has a length 1 with a(k, 1) = A099377(m) = A001600(k).

			The first ten rows of the triangle are:
  1,
  1, 3,
  1, 2,
  1, 1, 2, 2,
  1, 1, 2,
  2,
  1, 1, 3,
  2, 7, 2,
  2, 13,
  2, 4, 2
  ...

Cf. A001599, A001600, A099377, A099378.

Cf. A349474 (row lengths).

row[n_] := ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Table[row[k], {k, 1, 29}] // Flatten

A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 24, 25, 26, 27, 29, 31, 35, 37, 39, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 60, 61, 70, 73, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 105, 106, 107, 108, 110, 114, 115

Offset: 1

R. K. Guy, Feb 08 2004

nonn

The equation n = m*tau(m)/sigma(m) has an integer solution m.
Here tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).
A001600 sorted in order.
The Mersenne exponents (A000043) are in this sequence because the even perfect numbers, 2^(p-1)*(2^p-1) where p is in A000043, are all harmonic numbers (A001599) with harmonic mean of divisors p. - Amiram Eldar, Apr 15 2024

For further references see A001599.

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

Values of m are in A091911.
Complement of A157849.
Cf. A000043, A001599, A001600, A090758, A090759, A090760, A090761, A090762.

f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[ 0, {120}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 121 && a[[b]] == 0, a[[b]] = n], {n, 1, 560000000}]; Select[ Range[120], a[[ # ]] > 0 &] (* Robert G. Wilson v, Feb 14 2004 *)

More terms from Robert G. Wilson v, Feb 14 2004

cp's OEIS Frontend

A090758 Numbers k which occur exactly twice in A001600.

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A090759 Numbers n which occur at least twice in A001600.

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A090760 Numbers n which occur exactly three times in A001600.

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A090761 Numbers n which occur at least three times in A001600.

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A090762 Numbers n which occur exactly 4 times in A001600.

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

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A057660 a(n) = Sum_{k=1..n} n/gcd(n,k).

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A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.

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