Divisor Sum of 343

Example of Divisor Sum of Power of Prime

$\map {\sigma_1} {343} = 400$

where $\sigma_1$ denotes the divisor sum function.

Proof

From Divisor Sum of Power of Prime:

$\map {\sigma_1} {p^k} = \dfrac {p^{k + 1} - 1} {p_i - 1}$

We have that:

$343 = 7^3$

Hence:

\(\ds \map {\sigma_1} {343}\)

\(=\)

\(\ds \frac {7^4 - 1} {7 - 1}\)

\(\ds \)

\(=\)

\(\ds \frac {2400} 6\)

\(\ds \)

\(=\)

\(\ds 400\)

\(\ds \)

\(=\)

\(\ds 20^2\)

Thus we have that:

$7^0 + 7^2 + 7^2 + 7^3 = 20^2$

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $400$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $400$