Functions of Number Theory - 27.3 Multiplicative Properties

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
27.3.E3	${\displaystyle{\displaystyle\phi\left(n\right)=n\prod_{p\mathbin{|}n}(1-p^{-1})}}$ \Eulertotientphi[]@{n} = n\prod_{p\divides n}(1-p^{-1})		phi(n) = n*product(1 - (p)^(- 1), p**n in - infinity)	EulerPhi[n] == n*Product[1 - (p)^(- 1), {p**n, - Infinity}, GenerateConditions->None]	Translation Error	Translation Error	-	-
27.3.E5	${\displaystyle{\displaystyle d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1+a_{r})}}$ \ndivisors[]@{n} = \prod_{r=1}^{\nprimesdiv@{n}}(1+a_{r})		numelems(Divisors(n)) = product(1 + a[r], r = 1..ifactor(n))	Error	Error	Missing Macro Error	-	-
27.3.E6	${\displaystyle{\displaystyle\sigma_{\alpha}\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}\frac{p^{\alpha(1+a_{r})}_{r}-1}{p^{\alpha}_{r}-1}}}$ \sumdivisors{\alpha}@{n} = \prod_{r=1}^{\nprimesdiv@{n}}\frac{p^{\alpha(1+a_{r})}_{r}-1}{p^{\alpha}_{r}-1}	${\displaystyle{\displaystyle\alpha\neq 0}}$	add(divisors(alpha)) = product(((p[r])^(alpha*(1 + a[r]))- 1)/((p[r])^(alpha)- 1), r = 1..ifactor(n))	Error	Failure	Missing Macro Error	Error	-
27.3.E8	${\displaystyle{\displaystyle\phi\left(m\right)\phi\left(n\right)=\phi\left(mn\right)\phi\left(\left(m,n\right)\right)/\left(m,n\right)}}$ \Eulertotientphi[]@{m}\Eulertotientphi[]@{n} = \Eulertotientphi[]@{mn}\Eulertotientphi[]@{\pgcd{m,n}}/\pgcd{m,n}		phi(m)*phi(n) = phi(m*n)*phi(gcd(m , n))/gcd(m , n)	EulerPhi[m]*EulerPhi[n] == EulerPhi[m*n]*EulerPhi[GCD[m , n]]/GCD[m , n]	Failure	Failure	Failed [2 / 9] Result: -1. Test Values: {m = 2, n = 2} Result: -2. Test Values: {m = 3, n = 3}	Successful [Tested: 9]