Divisor function: Difference between revisions

Revision as of 17:31, 8 January 2021

Revision as of 22:30, 11 March 2021

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:<math>\begin{align}

\sigma(n) &= \sigma(n-1)+\sigma(n-2)-\sigma(n-5)-\sigma(n-7)+\sigma(n-12)+\sigma(n-15)+ \cdots \\

&= \sum_{i\in\Z} (-1)^{i+1}\left(\sigma \left (n-\frac12 \left (3i^2-i \right ) \right )+\deltasigma \left (n, -\frac12 \left (3i^2-+i \right ) \right ) n\right)

\end{align}</math>

where we set <math>\sigma(0)=n</math> if it occurs and <math>\sigma(i)=0</math> for <math>i \leq 0,</math>, we use the [[Kronecker delta]] <math>\delta(\cdot,\cdot),</math> and <math>\tfrac12 \left (3i^2-i \right )</math> are the [[pentagonal numbers]]. Indeed, Euler proved this by logarithmic differentiation of the identity in his [[Pentagonal number theorem]].

For a non-square integer, ''n'', every divisor, ''d'', of ''n'' is paired with divisor ''n''/''d'' of ''n'' and <math>\sigma_{0}(n)</math> is even; for a square integer, one divisor (namely <math>\sqrt n</math>) is not paired with a distinct divisor and <math>\sigma_{0}(n)</math> is odd. Similarly, the number <math>\sigma_{1}(n)</math> is odd if and only if ''n'' is a square or twice a square.{{Citation needed|date=May 2015}}