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

1,2

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.

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A001599, A003601. Different from A007340.

For the associated harmonic means, see A102408.

Sequence in context: A131492 A276026 A259718 * A140798 A325022 A337689

Adjacent sequences: A090942 A090943 A090944 * A090946 A090947 A090948

Keyword

nonn

Author

N. J. A. Sloane, Feb 28 2004

Status

approved