cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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
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

Examples

			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.

References

  • 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.

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

More terms from Klaus Brockhaus, Sep 18 2001

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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

Crossrefs

Cf. A001599, A090240 (sorted values).

Programs

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

Extensions

More terms from Matthew Conroy, Jan 15 2006

A091911 a(n) = smallest number m such that m*tau(m)/sigma(m) = n, or 0 if no such m exists.

Original entry on oeis.org

1, 6, 28, 0, 140, 270, 8128, 672, 1638, 6200, 2970, 0, 105664, 18620, 8190, 0, 27846, 0, 117800, 0, 55860, 0, 0, 30240, 173600, 242060, 167400, 0, 237510, 0, 2305843008139952128, 0, 0, 0, 2229500, 0, 4358600, 0, 726180, 0, 2290260, 1089270
Offset: 1

Views

Author

Robert G. Wilson v and R. K. Guy, Feb 11 2004

Keywords

Comments

Assumes that A090240 is correct.

Crossrefs

The zeros occur at the complement of A090240.

Programs

  • Mathematica
    f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[0, {50}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 51 && a[[b]] == 0, a[[b]] = n], {n, 1, 10 ^7}]; a

A090758 Numbers k which occur exactly twice in A001600.

Original entry on oeis.org

5, 13, 15, 17, 19, 24, 29, 37, 44, 45, 46, 47, 61, 73, 75, 85, 86, 87, 99, 105, 107, 144, 145, 149, 171, 173, 186, 187, 197, 203, 205, 224, 245, 252, 270, 272, 276, 279, 282, 284
Offset: 1

Views

Author

R. K. Guy, Feb 08 2004

Keywords

Examples

			k = 61 is a term because m*tau(m)/sigma(m) = 61 if and only if m = 2^30 * 61 * (2^31-1) or m = 2^60 * (2^61-1). - _Jinyuan Wang_, Mar 05 2020

Links

Crossrefs

Cf. A001600, A090240.

A090759 Numbers n which occur at least twice in A001600.

Original entry on oeis.org

5, 13, 15, 17, 19, 24, 27, 29, 37, 44, 45, 46, 47, 49, 51, 53, 61, 73, 75, 85, 86, 87, 89, 91, 96, 97, 99, 101, 105, 107, 144, 145, 149, 168, 169, 171, 173, 176, 181, 184, 186, 187, 188, 189, 191, 195, 197, 203, 205, 224, 240, 245, 252, 264, 270, 272
Offset: 1

Views

Author

R. K. Guy, Feb 08 2004

Keywords

References

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

Crossrefs

Cf. A001600, A090240.

A090760 Numbers n which occur exactly three times in A001600.

Original entry on oeis.org

27, 49, 51, 53, 89, 91, 97, 101, 168, 169, 181, 193, 195, 240, 264
Offset: 1

Views

Author

R. K. Guy, Feb 08 2004

Keywords

References

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

Crossrefs

Cf. A001600, A090240.

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

Original entry on oeis.org

27, 49, 51, 53, 89, 91, 96, 97, 101, 168, 169, 176, 181, 184, 188, 189, 191, 195, 240, 264, 285
Offset: 1

Views

Author

R. K. Guy, Feb 08 2004

Keywords

References

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

Crossrefs

Cf. A001600, A090240.

A090762 Numbers n which occur exactly 4 times in A001600.

Original entry on oeis.org

96, 176, 184, 188, 189, 191, 285
Offset: 1

Views

Author

R. K. Guy, Feb 08 2004

Keywords

References

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

Crossrefs

Cf. A001600, A090240.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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