Divisor function: Difference between revisions

Revision as of 17:40, 22 December 2021

Revision as of 17:58, 22 December 2021

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\end{align}</math>

where <math>\sigma(0)=n</math> if it occurs and <math>\sigma(xi)=0</math> for <math>xi <\leq 0,</math> <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}}

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is true for an infinity of values of n, see {{OEIS2C|A005237}}.

==Series relations==