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.

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

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:

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

$\blacksquare$

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