Integers whose Tau value equals Cube Root

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Theorem

There are $3$ positive integers whose $\tau$ value equals its cube root:

\(\displaystyle 1 = 1^3\) \(:\) \(\displaystyle \map \tau 1 = 1\) $\tau$ of $1$
\(\displaystyle 21 \, 952 = 28^3\) \(:\) \(\displaystyle \map \tau {21 \, 952} = 28\) $\tau$ of $21 \, 952$
\(\displaystyle 64 \, 000 = 40^3\) \(:\) \(\displaystyle \map \tau {64 \, 000} = 40\) $\tau$ of $64 \, 000$

This sequence is A066693 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Suppose $N = \map \tau {N^3}$.

The case $N = 1$ is trivial.


Suppose $N$ is a prime power.

Write $N = p^n$.

By Tau of Power of Prime:

$N = \map \tau {p^{3 n} } = 3 n + 1$

By Bernoulli's Inequality:

$N = p^n \ge 1 + n \paren {p - 1}$

This gives us the inequality:

$3 n + 1 \ge 1 + n \paren {p - 1}$

which can be simplified to:

$3 \ge p - 1$

The only primes satisfying the inequality are $2$ and $3$.


We have:

$\map \tau {2^3} = 4 > 2^1$
$\map \tau {2^6} = 7 > 2^2$
$\map \tau {2^9} = 10 > 2^3$
$\map \tau {2^{3 n} } = 3 n + 1 < 2^n$ for $n > 3$
$\map \tau {3^3} = 4 > 3^1$
$\map \tau {3^{3 n} } = 3 n + 1 < 3^n$ for $n > 1$
$\map \tau {p^{3 n} } = 3 n + 1 < p^n$ for all $p > 3$


Hence no prime powers satisfy the property.


Note that Tau Function is Multiplicative.

To form an integer $N$ with our property, we must choose and multiply prime powers from the list above.


If we chose any $\tuple {p, n}$ with $\map \tau {p^{4 n} } < p^n$, we must choose $2^m$ or $3^1$ in order for equality to possibly hold.

If $\tuple {2, 1}$ was chosen, $2^2 \nmid N$.

But $\map \tau {2^3} = 4 \divides N$, which is a contradiction.


Suppose $\tuple {2, 2}$ was chosen.

Then $\map \tau {2^6} = 7 \divides N$.

Then we must choose some $\tuple {7, n}$.

For $n = 1$, $\map \tau {7^3} = 4$.

$\map \tau {2^6 \times 7^3} = 4 \times 7 = 28 = 2^2 \times 3$
$\map \tau {2^6 \times 7^3 \times p^{3 m} } = 28 \paren {3 m + 1} < 28 \times p^m$ for all $p \ne 2, 7$

For $n > 1$, $\map \tau {2^6 \times 7^{3 n} } = 7 \paren {3 n + 1} < 4 \times 7^n$, a contradiction.


Suppose $\tuple {2, 3}$ was chosen.

Then:

$\map \tau {2^9} = 10 \divides N$

Then we must choose some $\tuple {5, n}$.

For $n = 1$, $\map \tau {5^3} = 4$.

$\map \tau {2^9 \times 5^3} = 10 \times 4 = 40 = 2^3 \times 5$
$\map \tau {2^9 \times 5^3 \times p^{3 m} } = 40 \paren {3 m + 1} < 40 \times p^m$ for all $p \ne 2, 5$

For $n > 1$, $\map \tau {2^9 \times 5^{3 n} } = 10 \paren {3 n + 1} < 2^3 \times 5^n$, a contradiction.


Suppose $\tuple {3, 1}$ was chosen.

Then $\map \tau {3^3} = 4 \divides N$.

Then this case coincides the cases above.


Thus we have exhausted all cases.

$\blacksquare$


Sources