%I A001599 M4185 N1743
%S A001599 1,6,28,140,270,496,672,1638,2970,6200,8128,8190,18600,18620,27846,30240,
%T A001599 32760,55860,105664,117800,167400,173600,237510,242060,332640,360360,
%U A001599 539400,695520,726180,753480,950976,1089270,1421280,1539720
%N A001599 Harmonic or Ore numbers: numbers n such that harmonic mean of divisors 
               of n is an integer.
%C A001599 Note that the harmonic mean of the divisors of n = n*tau(n)/sigma(n).
%C A001599 Equivalently, n*tau(n)/sigma(n) is an integer, where tau(n) (A000005) 
               is the number of divisors of n and sigma(n) is the sum of the divisors 
               of n (A000203).
%C A001599 Equivalently, the average of the divisors of n divides n.
%C A001599 Ore showed that every perfect number (A000396) is harmonic. The converse 
               does not hold: 140 is harmonic but not perfect. Ore conjectured that 
               1 is the only odd harmonic number.
%C A001599 Other example of power mean numbers n such that power mean of divisors 
               of n is an integer are RMS numbers A140480. [From Ctibor O.Zizka 
               (c.zizka(AT)email.cz), Sep 20 2008]
%D A001599 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A001599 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001599 G. L. Cohen and Deng Moujie, On a generalization of Ore's harmonic numbers, 
               Nieuw Arch. Wisk. (4), 16 (1998) 161-172.
%D A001599 G. L. Cohen and R. M. Sorli, Harmonic seeds, Fibonacci Quart., 36 (1998) 
               386-390; errata, 39 (2001) 4.
%D A001599 M. Garcia, On numbers with integral harmonic mean. Amer. Math. Monthly 
               61, (1954). 89-96.
%D A001599 T. Goto and S. Shibata, All numbers whose positive divisors have integral 
               harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.
%D A001599 R. K. Guy, Unsolved Problems in Number Theory, B2.
%D A001599 H.-J. Kanold, Uber das harmonische Mittel der Teiler einer natuerlichen 
               Zahl, Math. Ann., 133 (1957) 371-374.
%D A001599 W. H. Mills, On a conjecture of Ore, Proc. Number Theory Conf., Boulder 
               CO, 1972, 142-146.
%D A001599 O. Ore, On the averages of the divisors of a number, Amer. Math. Monthly, 
               55 (1948), 615-619.
%D A001599 Carl Pomerance, On a problem of Ore: Harmonic numbers (unpublished typescript); 
               see Abstract *709-A5, Notices Amer. Math. Soc., 20 (1973) Abstract 
               A-648.
%D A001599 Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. 
               Soc. Math. Grece (N.S.), 13 (1972) 12-22.
%H A001599 T. D. Noe and Klaus Brockhaus, <a href="b001599.txt">Table of n, a(n) 
               for n=1..170</a>
%H A001599 Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/html/harmonic-e.html#mark1">
               All harmonic numbers less than 10^14</a>
%H A001599 Takeshi Goto, <a href="http://www.ma.noda.tus.ac.jp/u/tg/files/list4">
               Table of a(n) for n=1..937</a>
%H A001599 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               HarmonicDivisorNumber.html">Harmonic Divisor Number</a>
%e A001599 n=140 has Sigma[ 0,140 ]=12 divisors with Sigma[ 1,140 ]=336. Average 
               divisor is 336/12=28, an integer and divides n: n=5*28. n=496, Sigma[ 
               0,496 ]=10, Sigma[ 1,496 ]=992: average divisor 99.2 is not an integer, 
               but n/(Sigma_1/Sigma_0)=496/99.2=5 is an integer.
%t A001599 Do[ If[ IntegerQ[ n*DivisorSigma[0, n]/ DivisorSigma[1, n]], Print[n]], 
               {n, 1, 1550000}]
%o A001599 (PARI) a(n)=if(n<0,0,n=a(n-1);until(0==(sigma(n,0)*n)%sigma(n,1),n++);
               n) - Michael Somos Feb 06 2004
%Y A001599 See A003601 for analogues referring to arithmetic mean and A000290 for 
               geometric mean of divisors.
%Y A001599 See A001600 and A090240 for the integer values obtained.
%Y A001599 sigma_0(n) (or tau(n)) is the number of divisors of n (A000005).
%Y A001599 sigma_1(n) (or sigma(n)) is the sum of the divisors of n (A000203).
%Y A001599 Cf. A007340, A090945, A035527, A007691, A074247, A053783. Not a subset 
               of A003601.
%Y A001599 Sequence in context: A084778 A155588 A108051 this_sequence A074247 A053783 
               A110047
%Y A001599 Adjacent sequences: A001596 A001597 A001598 this_sequence A001600 A001601 
               A001602
%K A001599 nonn,nice
%O A001599 1,2
%A A001599 N. J. A. Sloane (njas(AT)research.att.com).
%E A001599 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Sep 
               18 2001