A090945 Harmonic numbers (A001599) which are not perfect (A000396).

1, 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, 950976, 1089270, 1421280, 1539720

Offset: 1

Author

N. J. A. Sloane, Feb 28 2004

Keywords

• nonn

Examples

			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
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..930 (terms below 10^14; terms 1..77 from Muniru A Asiru)
• T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Crossrefs

Cf. A001599, A003601. Different from A007340.

For the associated harmonic means, see A102408.

Programs

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

Mathematica

Select[Range[2 10^7], IntegerQ[HarmonicMean[Divisors[#]]] && !DivisorSigma[1, #]==2 # &] (* Vincenzo Librandi, Nov 27 2018 *)

PARI

isok(n) = my(sn = sigma(n)); (frac(n*numdiv(n)/sn) == 0) && (sn != 2*n); \\ Michel Marcus, Nov 28 2018