cp's OEIS Frontend

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

%I A007340 M4299 #69 Apr 19 2022 07:26:50
%S A007340 1,6,140,270,672,1638,2970,6200,8190,18600,18620,27846,30240,32760,
%T A007340 55860,105664,117800,167400,173600,237510,242060,332640,360360,539400,
%U A007340 695520,726180,753480,1089270,1421280,1539720,2229500,2290260,2457000
%N A007340 Numbers whose divisors' harmonic and arithmetic means are both integers.
%C A007340 Intersection of A001599 and A003601.
%C A007340 The following are also in A046985: 1, 6, 672, 30240, 32760. Also contains multiply perfect (A007691) numbers. - _Labos Elemer_
%C A007340 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 A007340 Harmonic numbers k whose harmonic mean of divisors (A001600) is also a divisor of k. - _Amiram Eldar_, Apr 19 2022
%D A007340 G. L. Cohen, personal communication.
%D A007340 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
%D A007340 N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.
%D A007340 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007340 D. Wells, Curious and interesting numbers, Penguin Books, p. 124.
%H A007340 Donovan Johnson, <a href="/A007340/b007340.txt">Table of n, a(n) for n = 1..847</a>
%H A007340 G. L. Cohen, <a href="/A007340/a007340.pdf">Email to N. J. A. Sloane, Apr. 1994</a>
%H A007340 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 A007340 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha138.htm">Factorizations of many number sequences</a>
%H A007340 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 A007340 a = Sigma(1, x)/Sigma(0, x) integer and b = x/a also.
%e A007340 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 A007340 filter:= proc(n)
%p A007340 uses numtheory;
%p A007340 local a;
%p A007340 a:= sigma(n)/sigma[0](n);
%p A007340 type(a,integer) and type(n/a,integer);
%p A007340 end proc:
%p A007340 select(filter, [$1..2500000]); # _Robert Israel_, Oct 26 2014
%t A007340 Do[ a = DivisorSigma[0, n]/ DivisorSigma[1, n]; If[IntegerQ[n*a] && IntegerQ[1/a], Print[n]], {n, 1, 2500000}] (* _Labos Elemer_ *)
%t A007340 ahmQ[n_] := Module[{dn = Divisors[n]}, IntegerQ[Mean[dn]] && IntegerQ[HarmonicMean[dn]]]; Select[Range[2500000], ahmQ] (* _Harvey P. Dale_, Nov 16 2011 *)
%o A007340 (Haskell)
%o A007340 a007340 n = a007340_list !! (n-1)
%o A007340 a007340_list = filter ((== 0) . a054025) a001599_list
%o A007340 -- _Reinhard Zumkeller_, Dec 31 2013
%o A007340 (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 A007340 Intersection of A003601 and A001599.
%Y A007340 Different from A090945.
%Y A007340 Cf. A001600, A007691, A046985-A046987, A046999, A054025.
%K A007340 nonn,nice
%O A007340 1,2
%A A007340 _N. J. A. Sloane_
%E A007340 More terms from _Robert G. Wilson v_, Oct 03 2002
%E A007340 Edited by _N. J. A. Sloane_, Oct 05 2008 at the suggestion of _R. J. Mathar_