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

Author

R. K. Guy, Feb 08 2004

Keywords

• nonn

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

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

Crossrefs

Values of m are in A091911.

Complement of A157849.

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

Programs

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 *)

Extensions

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