%I #23 Oct 19 2024 08:34:49
%S 1,140,270,672,1638,2970,6200,8190,18600,18620,27846,30240,32760,
%T 55860,105664,117800,167400,173600,237510,242060,332640,360360,539400,
%U 695520,726180,753480,950976,1089270,1421280,1539720
%N Harmonic numbers (A001599) which are not perfect (A000396).
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
%H Amiram Eldar, <a href="/A090945/b090945.txt">Table of n, a(n) for n = 1..930</a> (terms below 10^14; terms 1..77 from Muniru A Asiru)
%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.
%e A001599(4) = 140, but 336 = sigma(140) <> 2*140 = 280. Thus, 140 is a harmonic number which is not perfect. - _Muniru A Asiru_, Nov 26 2018
%t Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* _Vincenzo Librandi_, Nov 27 2018 *)
%o (GAP) Concatenation([1],Filtered([2,4..2000000],n->Sigma(n)<>2*n and IsInt(n*Tau(n)/Sigma(n)))); # _Muniru A Asiru_, Nov 26 2018
%o (PARI) isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ _Michel Marcus_, Nov 28 2018
%Y Cf. A001599, A003601. Different from A007340.
%Y For the associated harmonic means, see A102408.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Feb 28 2004