Divisor function: Difference between revisions

Revision as of 01:56, 10 March 2022

Revision as of 02:01, 10 March 2022

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\end{align}</math>

where <math>\sigma(0)=n</math> if it occurs and <math>\sigma(ix)=0</math> for <math>ix \leq< 0</math>, and <math>\tfrac{1}{2} \left( 3i^2 \mp i \right)</math> are consecutive pairs of generalized [[pentagonal numbers]] ({{OEIS2C|A001318}}, starting at offset 1). 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.{{sfnp|Gioia|Vaidya|1967}}