A007340 - OEIS

%I M4299 #69 Apr 19 2022 07:26:50

%S 1,6,140,270,672,1638,2970,6200,8190,18600,18620,27846,30240,32760,

%T 55860,105664,117800,167400,173600,237510,242060,332640,360360,539400,

%U 695520,726180,753480,1089270,1421280,1539720,2229500,2290260,2457000

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

%C Intersection of A001599 and A003601.

%C The following are also in A046985: 1, 6, 672, 30240, 32760. Also contains multiply perfect (A007691) numbers. - _Labos Elemer_

%C 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

%C Harmonic numbers k whose harmonic mean of divisors (A001600) is also a divisor of k. - _Amiram Eldar_, Apr 19 2022

%D G. L. Cohen, personal communication.

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.

%D N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D D. Wells, Curious and interesting numbers, Penguin Books, p. 124.

%H Donovan Johnson, <a href="/A007340/b007340.txt">Table of n, a(n) for n = 1..847</a>

%H G. L. Cohen, <a href="/A007340/a007340.pdf">Email to N. J. A. Sloane, Apr. 1994</a>

%H T. Goto and S. Shibata, <a href="http://dx.doi.org/10.1090/S0025-5718-03-01554-0">All numbers whose positive divisors have integral harmonic mean up to 300</a>, Math. Comput. 73 (2004), 475-491.

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha138.htm">Factorizations of many number sequences</a>

%H Oystein Ore, <a href="http://www.jstor.org/stable/2305616">On the averages of the divisors of a number</a>, Amer. Math. Monthly, 55 (1948), 615-619.

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

%e 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.

%p filter:= proc(n)

%p uses numtheory;

%p local a;

%p a:= sigma(n)/sigma[0](n);

%p type(a,integer) and type(n/a,integer);

%p end proc:

%p select(filter, [$1..2500000]); # _Robert Israel_, Oct 26 2014

%t Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}] (* _Labos Elemer_ *)

%t ahmQ[n_] := Module[{dn = Divisors[n]}, IntegerQ[Mean[dn]] && IntegerQ[HarmonicMean[dn]]]; Select[Range[2500000], ahmQ] (* _Harvey P. Dale_, Nov 16 2011 *)

%o (Haskell)

%o a007340 n = a007340_list !! (n-1)

%o a007340_list = filter ((== 0) . a054025) a001599_list

%o -- _Reinhard Zumkeller_, Dec 31 2013

%o (PARI) is(n)=my(d=divisors(n),s=vecsum(d)); s%#d==0 && #d*n%s==0 \\ _Charles R Greathouse IV_, Feb 07 2017

%Y Intersection of A003601 and A001599.

%Y Different from A090945.

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

%K nonn,nice

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Robert G. Wilson v_, Oct 03 2002

%E Edited by _N. J. A. Sloane_, Oct 05 2008 at the suggestion of _R. J. Mathar_