A007340 -id:A007340 - 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

A003601 Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 105

Offset: 1

N. J. A. Sloane, Mira Bernstein

Sometimes called arithmetic numbers.
Generalized (sigma_r)-numbers are numbers j for which sigma_r(j)/sigma_0(j) = c^r. Sigma_r(j) denotes the sum of the r-th powers of the divisors of j; c,r are positive integers. The numbers in this sequence are sigma_1-numbers; those in A140480 are sigma_2-numbers. - Ctibor O. Zizka, Jul 14 2008
{a(n)} = union A175678 and A175679 where A175678 = numbers m such that the arithmetic mean Ad(m) of divisors of m and the arithmetic mean Ah(m) of numbers h < m such that gcd(h,m) = 1 are both integers and A175679 = numbers m such that the arithmetic mean Ad(m) of the divisors of m and the arithmetic mean Ak(m) of the numbers k <= m are both integers. - Jaroslav Krizek, Aug 07 2010
All odd primes (A065091) are arithmetic numbers. - Wesley Ivan Hurt, Oct 04 2013
A069928(n) = number of arithmetic numbers not greater than n. - Reinhard Zumkeller, Jul 28 2014
A102187(n) divides a(n) for a(n) = 1, 6, 140, 270, 672, ... A007340. - Thomas Ordowski, Oct 24 2014
The quotients sigma(j)/tau(j) are in A102187. - Bernard Schott, Jun 07 2017

			Sigma(6) = 12, tau(6) = 4, sigma(6)/tau(6) = 3 so 6 belongs to this sequence. - _Bernard Schott_, Jun 07 2017

R. K. Guy, Unsolved Problems in Number Theory, B2.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016.
Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer (1981). In Knopp, M.I. ed., Analytic number theory, Proc. Conf., Temple Univ., 1980. Lecture Notes in Mathematics. 899. Springer-Verlag., pp. 197-220.
Antonio M. Oller-Marcén, On arithmetic numbers, arXiv:1206.1823 [math.NT], 2012.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
Wikipedia, Arithmetic number.

Complement is A049642.
Cf. A000005, A000203, A054025, A001599, A007340, A140480, A102187.
Cf. A245644, A245656, A069928. Nonprimes are in A023883.

a:=Filtered([1..110],n->Sigma(n) mod Tau(n)=0);; Print(a); # Muniru A Asiru, Jan 25 2019

a003601 n = a003601_list !! (n-1)
a003601_list = filter ((== 1) . a245656) [1..]
-- Reinhard Zumkeller, Jul 28 2014, Dec 31 2013, Jan 06 2012

with(numtheory); t := [ ]: f := [ ]: for n from 1 to 500 do if sigma(n) mod tau(n) = 0 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # corrected by Wesley Ivan Hurt, Oct 03 2013

Select[Range[120], IntegerQ[DivisorSigma[1, # ]/DivisorSigma[0, # ]] &] (* Stefan Steinerberger, Apr 03 2006 *)

is(n)=sigma(n)%numdiv(n)==0 \\ Charles R Greathouse IV, Jul 10 2012

from sympy import divisors, divisor_count
[n for n in range(1,10**5) if not sum(divisors(n)) % divisor_count(n)] # Chai Wah Wu, Aug 05 2014

a(n) ~ n. - Charles R Greathouse IV, Jul 10 2012

A245656(a(n)) = 1. - Reinhard Zumkeller, Jul 28 2014

David W. Wilson, Oct 15 1996, points out that 30 was missing.

More terms from Stefan Steinerberger, Apr 03 2006

A002387 Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

Original entry on oeis.org

1, 2, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422

Offset: 0

N. J. A. Sloane

From Dean Hickerson, Apr 19 2003: (Start)
For k >= 1, log(k + 1/2) + gamma < H(k) < log(k + 1/2) + gamma + 1/(24k^2), where gamma is Euler's constant (A001620). It is likely that the upper and lower bounds have the same floor for all k >= 2, in which case a(n) = floor(exp(n-gamma) + 1/2) for all n >= 0.
This remark is based on a simple heuristic argument. The lower and upper bounds differ by 1/(24k^2), so the probability that there's an integer between the two bounds is 1/(24k^2). Summing that over all k >= 2 gives the expected number of values of k for which there's an integer between the bounds. That sum equals Pi^2/144 - 1/24 ~ 0.02687. That's much less than 1, so it is unlikely that there are any such values of k.
(End)
Referring to A118050 and A118051, using a few terms of the asymptotic series for the inverse of H(x), we can get an expression which, with greater likelihood than mentioned above, should give a(n) for all n >= 0. For example, using the same type of heuristic argument given by Dean Hickerson, it can be shown that, with probability > 99.995%, we should have, for all n >= 0, a(n) = floor(u + 1/2 - 1/(24u) + 3/(640u^3)) where u = e^(n - gamma). - David W. Cantrell (DWCantrell(AT)sigmaxi.net)
For k > 1, H(k) is never an integer. Hence apart from the first two terms this sequence coincides with A004080. - Nick Hobson, Nov 25 2006

John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, pages 258-259.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 28, Ellipses, Paris 2008.
Ronald Lewis Graham, Donald Ervin Knuth and Oren Patashnik, "Concrete Mathematics, a Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, Page 258-264, 438.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 23 1973.
W. Sierpiński, Sur les decompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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).
I. Stewart, L'univers des nombres, pp. 54, Belin-Pour La Science, Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 0..2303 (first 101 terms from T. D. Noe)
John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313. (Gives terms up to n = 24.)
R. P. Boas, Partial sums of the harmonic series, II, Mimeographed manuscript, no date.
R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870. (Gives terms up to n = 20.)
Nick Hobson, Solution to puzzle 34: Harmonic sum 2.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 28 1973.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
H. P. Robinson, Letter to N. J. A. Sloane, Dec 20 1983
John A. Rochowicz, Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE) (2015), Vol. 8: Iss. 2, Article 4.
R. G. Wilson, Letter to N. J. A. Sloane with attachment, Jan 1989 (A006509 is mentioned in the attachment)
R. G. Wilson v, Letter to N. J. A. Sloane, Oct 12 1993
J. W. Wrench, Jr., Selected Partial Sums of the Harmonic Series, Manuscript, no date [Annotated scanned copy]

Apart from initial terms, same as A004080.

Cf. A006509, A055980, A115515, A242654.

a002387 n = a002387_list !! n
a002387_list = f 0 1 where
   f x k = if hs !! k > fromIntegral x
           then k : f (x + 1) (k + 1) else f x (k + 1)
           where hs = scanl (+) 0 $ map recip [1..]
-- Reinhard Zumkeller, Aug 04 2014

fh[0]=0; fh[1]=1; fh[k_] := Module[{tmp}, If[Floor[tmp=Log[k+1/2]+EulerGamma]==Floor[tmp+1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0]=1; a[1]=2; a[n_] := Module[{val}, val=Round[Exp[n-EulerGamma]]; If[fh[val]==n&&fh[val-1]==n-1, val, UNKNOWN]]; (* fh[k] is either floor(H(k)) or UNKNOWN *)
f[n_] := k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100] // Ceiling; f[0] = 1; Array[f, 28, 0] (* Robert G. Wilson v, Jan 24 2017 after Jean-François Alcover in A014537 *)

a(n)=if(n,my(k=exp(n-Euler));ceil(solve(x=k-1.5,k+.5,intnum(y=0,1,(1-y^x)/(1-y))-n)),1) \\ Charles R Greathouse IV, Jun 13 2012

Note that the conditionally convergent series Sum_{k>=1} (-1)^(k+1)/k = log 2 (A002162).
Limit_{n->oo} a(n+1)/a(n) = e. - Robert G. Wilson v, Dec 07 2001
It is conjectured that, for n > 1, a(n) = floor(exp(n-gamma) + 1/2). - Benoit Cloitre, Oct 23 2002

Terms for n >= 13 computed by Eric W. Weisstein; corrected by James R. Buddenhagen and Eric W. Weisstein, Feb 18 2001

Edited by Dean Hickerson, Apr 19 2003

A004080 Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

Original entry on oeis.org

0, 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422

Offset: 0

N. J. A. Sloane, Clark Kimberling

			a(2)=4 because 1/1 + 1/2 + 1/3 + 1/4 > 2.

Bruno Rizzi and Cristina Scagliarini: I numeri armonici. Periodico di matematiche, "Mathesis", pp. 17-58, 1986, numbers 1-2. [From Vincenzo Librandi, Jan 05 2009]
W. Sierpiński, Sur les décompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.
N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

T. D. Noe, Table of n, a(n) for n=0..100 (using Hickerson's formula in A002387)
John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313.
R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870.
Keneth Adrian Dagal, A Lower Bound for tau(n) for k-Multiperfect Number, arXiv:1309.3527 [math.NT], 2013.
J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number
Eric Weisstein's World of Mathematics, Harmonic Series
Eric Weisstein's World of Mathematics, High-Water Mark

Apart from first two terms, same as A002387.

import Data.List (findIndex); import Data.Maybe (fromJust)
a004080 n = fromJust $
   findIndex (fromIntegral n <=) $ scanl (+) 0 $ map recip [1..]
-- Reinhard Zumkeller, Jul 13 2014

aux[0] = 0; Do[aux[n] = Floor[Floor[Sum[1/i, {i, n}]]]; If[aux[n] > aux[n - 1], Print[n]], {n, 1, 14000}] (* José María Grau Ribas, Feb 20 2010 *)
a[0] = 0; a[1] = 1; a[n_] := k /. FindRoot[ HarmonicNumber[k] == n, {k, Exp[n - EulerGamma]}, WorkingPrecision -> 50] // Ceiling; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 13 2013, after Charles R Greathouse IV *)

my(t=0, n=0); for(i=0, 10^20, if (i, t+=1./i); if(t>=n, print1(i, ", "); n++)) \\ Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007; corrected by Michel Marcus, Jan 19 2022

Limit_{n->oo} a(n+1)/a(n) = exp(1). - Sébastien Dumortier, Jun 29 2005
a(n) = exp(n - gamma + o(1)). - Charles R Greathouse IV, Mar 10 2009
a(n) = A002387(n) for n>1. - Robert G. Wilson v, Jun 18 2015

Terms for n >= 13 computed by Eric W. Weisstein; corrected by James R. Buddenhagen and Eric W. Weisstein, Feb 18 2001
Edited by Dean Hickerson, Apr 19 2003
More terms from Sébastien Dumortier, Jun 29 2005
a(27) from Thomas Gettys (tpgettys(AT)comcast.net), Dec 05 2006
a(28) from Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007
Edited by Charles R Greathouse IV, Mar 24 2010

A001600 Harmonic means of divisors of harmonic numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 5, 8, 9, 11, 10, 7, 15, 15, 14, 17, 24, 24, 21, 13, 19, 27, 25, 29, 26, 44, 44, 29, 46, 39, 46, 27, 42, 47, 47, 54, 35, 41, 60, 51, 37, 48, 45, 49, 50, 49, 53, 77, 86, 86, 51, 96, 75, 70, 80, 99, 110, 81, 84, 13, 102, 82, 96, 114, 53, 108, 115, 105, 116, 91, 85, 105

Offset: 1

N. J. A. Sloane

Values of n*tau(n)/sigma(n) corresponding to terms of A001599, where tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).

Kanold (1957) proved that each term appears only a finite number of times. - Amiram Eldar, Jun 01 2020

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).

R. J. Mathar, Table of n, a(n) for n = 1..937, extending the former b-file of T. D. Noe.
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
M. Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly 61, (1954). 89-96.
Takeshi Goto, All harmonic numbers less than 10^14
Takeshi Goto, Table of a(n) for n=1..937
Hans-Joachim Kanold , Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann., Vol. 133 (1957), pp. 371-374.
O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.
O. Ore, On the averages of the divisors of a number (annotated scanned copy)

Cf. A001599, A090240 (sorted values).

a001600 n = a001600_list !! (n-1)
a001600_list =
   [numerator m | x <- [1..], let m = hm x, denominator m == 1] where
   hm x = genericLength divs * recip (sum $ map recip divs)
          where divs = map fromIntegral $ a027750_row x
-- Reinhard Zumkeller, Apr 01 2014

A001600 = Reap[Do[tau = DivisorSigma[0, n]; sigma = DivisorSigma[1, n]; h = n*tau/sigma; If[IntegerQ[h], Print[h]; Sow[h]], {n, 1, 90000000}]][[2, 1]](* Jean-François Alcover, May 11 2012 *)

lista(nn) = for (n=1, nn, if (denominator(q=n*numdiv(n)/sigma(n)) == 1, print1(q, ", "))); \\ Michel Marcus, Jan 13 2016

More terms from Matthew Conroy, Jan 15 2006

A090945 Harmonic numbers (A001599) which are not perfect (A000396).

Original entry on oeis.org

1, 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, 950976, 1089270, 1421280, 1539720

Offset: 1

N. J. A. Sloane, Feb 28 2004

nonn

			A001599(4) = 140, but 336 = sigma(140) <> 2*140 = 280. Thus, 140 is a harmonic number which is not perfect. - _Muniru A Asiru_, Nov 26 2018

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.

Amiram Eldar, Table of n, a(n) for n = 1..930 (terms below 10^14; terms 1..77 from Muniru A Asiru)
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Cf. A001599, A003601. Different from A007340.

For the associated harmonic means, see A102408.

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

Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* Vincenzo Librandi, Nov 27 2018 *)

isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ Michel Marcus, Nov 28 2018

A065071 Minimum number of identical bricks of length 1 which, when stacked without mortar in the naive way, form a stack of length >=n.

Original entry on oeis.org

1, 5, 32, 228, 1675, 12368, 91381, 675215, 4989192, 36865413, 272400601, 2012783316, 14872568832, 109894245430, 812014744423, 6000022499694, 44334502845081, 327590128640501, 2420581837980562, 17885814992891027

Offset: 1

John W. Layman, Nov 08 2001

Note that one can do "better" in terms of projections if one groups the bricks asymmetrically into lozenges with holes. See the Ainsley and Drummond references. Ainsley considers only the case of four bricks, but achieves an overhang of (15 - 4*sqrt(2))/8, compared with 25/24 for the harmonic pile. - D. G. Rogers, Aug 31 2005

Lim_{n -> inf} a(n)/a(n-1) = exp(2). - Robert G. Wilson v, Jan 26 2017

			Obviously a(1)=1. If the center of gravity of one brick is placed at the end of a second brick, the length of the stack of 2 bricks is 1.5. If the c.g. of that stack is placed at the end of a third brick, the length of the stack is 1.75. Continuing, we get a stack of length 1.916666... for 4 bricks and a stack of length 2.0416666... for 5 bricks. Thus a(2)=5.

N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
S. Ainley, Finely Balanced, Math. Gaz., 63 (1979), 272.
R. Dickau, Harmonic numbers and the book-stacking problem
J. E. Drummond, On stacking bricks to achieve a large overhang, Math. Gaz., 65 (1981), 40-42.
Eric Weisstein's World of Mathematics, Book Stacking Problem

Cf. harmonic numbers H(n) = A001008/A002805, A002387, A004080.

A002387[n_] := Floor[ Exp[n - EulerGamma] + 1/2]; a[n_] := A002387[2n - 2] + 1; a[1] = 1; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Dec 13 2011, after Charles R Greathouse IV *)
f[n_] := k /. FindRoot[HarmonicNumber[k -1] == 2n, {k, Exp[ 2n]}, WorkingPrecision -> 100] // Ceiling; Array[f, 21, 0] (* Robert G. Wilson v, Jan 26 2017 after Jean-François Alcover in A014537 *) (* note that the index is off by one *)

a(n) = A002387(2n) + 1 = A014537(n) + 1.

More terms from Vladeta Jovovic, Nov 14 2001

cp's OEIS Frontend

A157848 Corresponding values of arithmetic means of divisors of numbers from A007340.

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A090944 Duplicate of A007340.

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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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A003601 Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)).

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A002387 Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

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A004080 Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

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