Number of divisors function: Difference between revisions
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In [[number theory]] the '''number of divisors function''' of a positive integer, denoted ''d''(''n'') or τ(''n''), is the number of positive integer [[divisor]]s of the number ''n''. | In [[number theory]] the '''number of divisors function''' of a positive integer, denoted ''d''(''n'') or τ(''n'') or σ<sub>0</sub>(''n''), is the number of positive integer [[divisor]]s of the number ''n''. | ||
It is a [[multiplicative function]], that | It is a [[multiplicative function]], that is ''m'' and ''n'' are coprime then <math>d(mn) = d(m) d(n)</math>. | ||
The value of ''d'' on a general integer ''n'' with prime factorisation | The value of ''d'' on a general integer ''n'' with prime factorisation | ||
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The [[Average order of an arithmetic function|average order]] of ''d''(''n'') is <math>\log(n)</math>. | The [[Average order of an arithmetic function|average order]] of ''d''(''n'') is <math>\log(n)</math>. | ||
The [[Normal order of an arithmetic function|normal order]] of log(''d''(''n'')) is log(2) log log(''n''). | The [[Normal order of an arithmetic function|normal order]] of log(''d''(''n'')) is log(2) log log(''n'').[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 11:01, 27 September 2024
In number theory the number of divisors function of a positive integer, denoted d(n) or τ(n) or σ0(n), is the number of positive integer divisors of the number n.
It is a multiplicative function, that is m and n are coprime then .
The value of d on a general integer n with prime factorisation
is then
The average order of d(n) is . The normal order of log(d(n)) is log(2) log log(n).
