A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 24, 25, 26, 27, 29, 31, 35, 37, 39, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 60, 61, 70, 73, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 105, 106, 107, 108, 110, 114, 115

Offset

1,2

Comments

The equation n = m*tau(m)/sigma(m) has an integer solution m.

Here tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).

A001600 sorted in order.

The Mersenne exponents (A000043) are in this sequence because the even perfect numbers, 2^(p-1)*(2^p-1) where p is in A000043, are all harmonic numbers (A001599) with harmonic mean of divisors p. - Amiram Eldar, Apr 15 2024

References

For further references see A001599.

Links

Table of n, a(n) for n=1..69.

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Mathematica

f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[ 0, {120}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 121 && a[[b]] == 0, a[[b]] = n], {n, 1, 560000000}]; Select[ Range[120], a[[ # ]] > 0 &] (* Robert G. Wilson v, Feb 14 2004 *)

Crossrefs

Values of m are in A091911.

Complement of A157849.

Cf. A000043, A001599, A001600, A090758, A090759, A090760, A090761, A090762.

Sequence in context: A119605 A144146 A284763 * A137407 A261406 A264387

Adjacent sequences: A090237 A090238 A090239 * A090241 A090242 A090243

Keyword

nonn

Author

R. K. Guy, Feb 08 2004

Extensions

More terms from Robert G. Wilson v, Feb 14 2004

Status

approved