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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 (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

REFERENCES

G. L. Cohen, personal communication.

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

R. K. Guy, Unsolved Problems in Number Theory, B2.

O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 55 (1948), 615-619.

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

Hisanori Mishima, Factorizations of many number sequences

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.

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

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

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Last modified July 26 03:04 EDT 2014. Contains 244924 sequences.