

A007340


Numbers n such that both the harmonic and arithmetic means of the divisors of n are integers.
(Formerly M4299)


10



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
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OFFSET

1,2


COMMENTS

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 having the average divisor. The Ore's harmonic numbers A001599 without the numbers A046999.  Thomas Ordowski, Oct 26 2014


REFERENCES

G. L. Cohen, personal communication.
R. K. Guy, Unsolved Problems in Number Theory, B2.
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.


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..847
T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475491.
Hisanori Mishima, Factorizations of many number sequences
Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615619.


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:
select(filter, [$1..2500000]); # Robert Israel, Oct 26 2014


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 !! (n1)
a007340_list = filter ((== 0) . a054025) a001599_list
 Reinhard Zumkeller, Dec 31 2013


CROSSREFS

Intersection of A003601 and A001599. Cf. A007691, A046985  A046987, A046999.
Different from A090945.
Cf. A054025.
Sequence in context: A155558 A053467 A090944 * A122483 A123729 A193835
Adjacent sequences: A007337 A007338 A007339 * A007341 A007342 A007343


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

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


STATUS

approved



