Functions of Number Theory - 27.3 Multiplicative Properties
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 27.3.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Eulertotientphi[]@{n} = n\prod_{p\divides n}(1-p^{-1})}
\Eulertotientphi[]@{n} = n\prod_{p\divides n}(1-p^{-1}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | phi(n) = n*product(1 - (p)^(- 1), p**n in - infinity)
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EulerPhi[n] == n*Product[1 - (p)^(- 1), {p**n, - Infinity}, GenerateConditions->None]
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Translation Error | Translation Error | - | - |
| 27.3.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ndivisors[]@{n} = \prod_{r=1}^{\nprimesdiv@{n}}(1+a_{r})}
\ndivisors[]@{n} = \prod_{r=1}^{\nprimesdiv@{n}}(1+a_{r}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | numelems(Divisors(n)) = product(1 + a[r], r = 1..ifactor(n))
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Error
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Error | Missing Macro Error | - | - |
| 27.3.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sumdivisors{\alpha}@{n} = \prod_{r=1}^{\nprimesdiv@{n}}\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} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha \neq 0} | add(divisors(alpha)) = product(((p[r])^(alpha*(1 + a[r]))- 1)/((p[r])^(alpha)- 1), r = 1..ifactor(n))
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Error
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Failure | Missing Macro Error | Error | - |
| 27.3.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Eulertotientphi[]@{m}\Eulertotientphi[]@{n} = \Eulertotientphi[]@{mn}\Eulertotientphi[]@{\pgcd{m,n}}/\pgcd{m,n}}
\Eulertotientphi[]@{m}\Eulertotientphi[]@{n} = \Eulertotientphi[]@{mn}\Eulertotientphi[]@{\pgcd{m,n}}/\pgcd{m,n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | phi(m)*phi(n) = phi(m*n)*phi(gcd(m , n))/gcd(m , n)
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EulerPhi[m]*EulerPhi[n] == EulerPhi[m*n]*EulerPhi[GCD[m , n]]/GCD[m , n]
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Failure | Failure | Failed [2 / 9] Result: -1.
Test Values: {m = 2, n = 2}
Result: -2.
Test Values: {m = 3, n = 3}
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Successful [Tested: 9] |