OFFSET
1,2
COMMENTS
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
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.
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 !! (n-1)
a007340_list = filter ((== 0) . a054025) a001599_list
-- Reinhard Zumkeller, Dec 31 2013
(PARI) is(n)=my(d=divisors(n), s=vecsum(d)); s%#d==0 && #d*n%s==0 \\ Charles R Greathouse IV, Feb 07 2017
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
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