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 A007340 Numbers n such that both the harmonic and arithmetic means of the divisors of n are integers. (Formerly M4299) 11
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 Intersection of A003601 and A001599. Different from A090945. Cf. A007691, A046985-A046987, A046999, A054025. Sequence in context: A288557 A288565 A090944 * A122483 A123729 A193835 Adjacent sequences:  A007337 A007338 A007339 * A007341 A007342 A007343 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

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Last modified January 20 02:13 EST 2019. Contains 319320 sequences. (Running on oeis4.)