A140798 Harmonic numbers that are not multiply-perfect.

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 332640, 360360, 539400, 695520, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600

Offset: 1

Author

Lekraj Beedassy, Jul 15 2008

Keywords

• nonn

Comments

Sequence is A001599 excluding those entries that appear in A007691.

Multiply-perfect numbers m (with sigma(m)/m an integer) are necessarily harmonic numbers (with tau(m)/{sigma(m)/m } an integer), but the converse is not true : If m divides sigma(m), then quotient sigma(m)/m divides tau(m) [m=A007691]; However, quotient tau(n)/{sigma(n)/n} being an integer does not imply quotient sigma(n)/n is necessarily an integer [n=A001599].

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 140, pp 48, Ellipses, Paris 2008.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..912 (terms < 10^14)

Crossrefs

Cf. A000203, A054030, A000005.

Programs

Mathematica

Select[Range[10^6], (d = Divisors[#]; IntegerQ[HarmonicMean[d]] && Mod[Total[d], #] != 0)&] (* Jean-François Alcover, Nov 13 2017 *)

Extensions

Initial term 1 removed and offset corrected by Donovan Johnson, Jan 25 2012