Divisor function: Difference between revisions

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==Definition==

The '''sum of positive divisors function''' σ_''xz''(''n''), for a real or complex number ''xz'', is defined as the [[summation|sum]] of the ''xz''th [[Exponentiation|powers]] of the positive [[divisor]]s of ''n''. It can be expressed in [[Summation#Capital-sigma notation|sigma notation]] as

:<math>\sigma_xsigma_z(n)=\sum_{d\mid n} d^xz\,\! ,</math>

where <math>{d\mid n}</math> is shorthand for "''d'' [[divides]] ''n''".

The notations ''d''(''n''), ν(''n'') and τ(''n'') (for the German ''Teiler'' = divisors) are also used to denote σ₀(''n''), or the '''number-of-divisors function'''<ref name="Long 1972 46">{{harvtxt|Long|1972|p=46}}</ref><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=63}}</ref> ({{OEIS2C|id=A000005}}). When ''xz'' is 1, the function is called the '''sigma function''' or '''sum-of-divisors function''',<ref name="Long 1972 46"/><ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=58}}</ref> and the subscript is often omitted, so σ(''n'') is the same as σ₁(''n'') ({{OEIS2C|id=A000203}}).

The '''[[aliquot sum]]''' ''s''(''n'') of ''n'' is the sum of the [[proper divisor]]s (that is, the divisors excluding ''n'' itself, {{OEIS2C|id=A001065}}), and equals σ₁(''n'')&nbsp;&minus;&nbsp;''n''; the [[aliquot sequence]] of ''n'' is formed by repeatedly applying the aliquot sum function.