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Harmonic means associated with A090945.
+20
1
1, 5, 6, 8, 9, 11, 10, 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, 102, 82, 96, 114, 53, 108, 115, 105, 116, 91
COMMENTS
A perfect number N=2^(p-1)*M(p) (= A000396), {with associated Mersenne prime M(p)=2^p -1 (= A000668)}, has harmonic mean N*tau(N)/sigma(N) = p (= A000043).
PROG
(PARI) lista(nn) = {for (n=1, nn, ab = sigma(n)/n; hm = numdiv(n)/ab; if ((type(hm) == "t_INT") && (ab != 2), print1(hm, ", ")); ); } \\ Michel Marcus, Oct 26 2013
EXTENSIONS
a(1)=1 as well as a(29)=47 inserted and more terms added by Michel Marcus, Oct 26 2013
Harmonic or Ore numbers: numbers k such that the harmonic mean of the divisors of k is an integer.
(Formerly M4185 N1743)
+10
114
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
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 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
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).
LINKS
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, The Biharmonic mean, arXiv:1601.03081 [math.NT], 2016. Graeme L. Cohen and Ronald M. Sorli, Harmonic seeds, Fibonacci Quart., Vol. 36, No. 5 (1998), pp. 386-390; errata, 39 (2001) 4. 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.
EXAMPLE
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.
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:
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 *)
PROG
(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 */ (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
(GAP) Concatenation([1], Filtered([2, 4..2000000], n->IsInt(n*Tau(n)/Sigma(n)))); # Muniru A Asiru, Nov 26 2018 (Python)
from sympy import divisor_sigma as sigma
def ok(n): return (n*sigma(n, 0))%sigma(n, 1) == 0
(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
CROSSREFS
See A003601 for analogs referring to arithmetic mean and A000290 for geometric mean of divisors. 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).
Numbers whose divisors' harmonic and arithmetic means are both integers.
(Formerly M4299)
+10
18
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
COMMENTS
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
REFERENCES
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.
FORMULA
a = Sigma(1, x)/Sigma(0, x) integer and b = x/a also.
EXAMPLE
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.
MAPLE
filter:= proc(n)
uses numtheory;
local a;
a:= sigma(n)/sigma[0](n);
type(a, integer) and type(n/a, integer);
end proc:
MATHEMATICA
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 *)
PROG
(Haskell)
a007340 n = a007340_list !! (n-1)
a007340_list = filter ((== 0) . a054025) a001599_list
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