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→Formulas at prime powers: Explain further an imporant result |
→Other properties and identities: Every square of a square is also a square, so ((square || square-of-a-square) <=> square), the "or" term does not expand the ___domain of the constraint. Looking for a missing citation...- Tag: Reverted |
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where <math>\sigma(0)=n</math> if it occurs and <math>\sigma(i)=0</math> for <math>i \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
We also note ''s''(''n'') = ''σ''(''n'') − ''n''. Here ''s''(''n'') denotes the sum of the proper divisors of ''n'', that is, the divisors of ''n'' excluding ''n'' itself. This function is the one used to recognize [[perfect number]]s which are the ''n'' for which ''s''(''n'') = ''n''. If ''s''(''n'') > ''n'' then ''n'' is an [[abundant number]] and if ''s''(''n'') < ''n'' then ''n'' is a [[deficient number]].
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