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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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 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

Examples

			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.
		

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.

Links

• 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.

Crossrefs

Intersection of A003601 and A001599.

Different from A090945.

Cf. A001600, A007691, A046985-A046987, A046999, A054025.

Programs

Haskell

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

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 *)

PARI

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

Formula

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

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