A number
n
>
1
n>1
is harmonic if
σ
(
n
)
∣
n
τ
(
n
)
\sigma (n)\mid n\tau (n)
, where
τ
(
n
)
\tau (n)
and
σ
(
n
)
\sigma (n)
are the number of positive divisors of
n
n
and their sum, respectively. It is known that there are no odd harmonic numbers up to
10
15
10^{15}
. We show here that, for any odd number
n
>
10
6
n>10^6
,
τ
(
n
)
≤
n
1
/
3
\tau (n)\le n^{1/3}
. It follows readily that if
n
n
is odd and harmonic, then
n
>
p
3
a
/
2
n>p^{3a/2}
for any prime power divisor
p
a
p^a
of
n
n
, and we have used this in showing that
n
>
10
18
n>10^{18}
. We subsequently showed that for any odd number
n
>
9
⋅
10
17
n>9\cdot 10^{17}
,
τ
(
n
)
≤
n
1
/
4
\tau (n)\le n^{1/4}
, from which it follows that if
n
n
is odd and harmonic, then
n
>
p
8
a
/
5
n>p^{8a/5}
with
p
a
p^a
as before, and we use this improved result in showing that
n
>
10
24
n>10^{24}
.