Euler's Pentagonal Numbers Theorem/Corollary 1/Examples/12

Example of Euler's Pentagonal Numbers Theorem: Corollary 1

$\map {\sigma_1} {12} = \map {\sigma_1} {11} + \map {\sigma_1} {10} - \map {\sigma_1} 7 - \map {\sigma_1} 5 + 12$

Proof

\(\ds \map {\sigma_1} {12}\)

\(=\)

\(\ds \sum_{1 \mathop \le 12 - GP_k \mathop < 12} -\paren {-1}^{\ceiling {k / 2} } \map {\sigma_1} {12 - GP_k} + 12 \sqbrk {\exists k \in \Z: GP_k = 12}\)

Euler's Pentagonal Numbers Theorem: Corollary 1

\(\ds \)

\(=\)

\(\ds \map {\sigma_1} {11} + \map {\sigma_1} {10} - \map {\sigma_1} 7 - \map {\sigma_1} 5 + 12\)

as $GP_5 = 12$

\(\ds \)

\(=\)

\(\ds 12 + 3 \times 6 - 8 - 6 + 12\)

Divisor Sum of Integer, Divisor Sum of Prime Number

\(\ds \)

\(=\)

\(\ds 28\)

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$