hcju/proofwikips huggingface dataset:
Euler Phi Function of 105
== Example of Euler $\phi$ Function of Square-Free Integer ==  :$\map \phi {105} = 48$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\displaystyle \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$ where $p \divides n$ denotes the primes which divide $n$. We have that: :$105 = 3 \times 5 \times 7$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \map \phi {105}       | r = \paren {3 - 1} \paren {5 - 1} \paren {7 - 1}       | c =  }} {{eqn | r = 2 \times 4 \times 6       | c =  }} {{eqn | r = 48       | c =  }} {{end-eqn}}
Euler Phi Function of 15
Euler Phi Function of 15 0 56579 297640 2017-05-16T20:46:52Z Prime.mover 59 Created page with "== Example of Euler $\phi$ Function of Non-Square Semiprime ==  :$\phi \left({15}\right) = 8$  where..." wikitext text/x-wiki == Example of Euler $\phi$ Function of Non-Square Semiprime ==  :$\phi \left({15}\right) = 8$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == We have that: :$15 = 3 \times 5$ Thus: {{begin-eqn}} {{eqn | l = \phi \left({15}\right)       | r = \left({3 - 1}\right) \left({5 - 1}\right)       | c = Euler $\phi$ Function of Non-Square Semiprime }} {{eqn | r = 2 \times 4       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}}
Divisor Counting Function/Examples/105
Divisor Counting Function/Examples/105 0 56595 451773 438890 2020-03-02T09:43:28Z Prime.mover 59 Prime.mover moved page Tau Function/Examples/105 to Divisor Counting Function/Examples/105 wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\map \tau {105} = 8$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$105 = 3 \times 5 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {105}       | r = \map \tau {3^1 \times 5^1 \times 7^1}       | c =  }} {{eqn | r = \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $105$ can be enumerated as: :$1, 3, 5, 7, 15, 21, 35, 105$ {{OEIS|A018286}}
Euler Phi Function of 165
Euler Phi Function of 165 0 56604 297726 2017-05-17T20:19:14Z Prime.mover 59 Created page with "== Example of Euler $\phi$ Function of Square-Free Integer ==  :$\phi \left({165}\right) = 80$  where..." wikitext text/x-wiki == Example of Euler $\phi$ Function of Square-Free Integer ==  :$\phi \left({165}\right) = 80$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\displaystyle \phi \left({n}\right) = \prod_{\substack {p \mathop \backslash n \\ p \mathop > 2} } \left({p - 1}\right)$ where $p \mathop \backslash n$ denotes the primes which divide $n$. We have that: :$165 = 3 \times 5 \times 11$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \phi \left({165}\right)       | r = \left({3 - 1}\right) \left({5 - 1}\right) \left({11 - 1}\right)       | c =  }} {{eqn | r = 2 \times 4 \times 10       | c =  }} {{eqn | r = 80       | c =  }} {{end-eqn}}
Euler Phi Function of 35
== Example of Euler $\phi$ Function of Non-Square Semiprime ==  :$\map \phi {35} = 24$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == We have that: :$35 = 5 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \map \phi {35}       | r = \paren {5 - 1} \paren {7 - 1}       | c = Euler $\phi$ Function of Non-Square Semiprime }} {{eqn | r = 4 \times 6       | c =  }} {{eqn | r = 24       | c =  }} {{end-eqn}}
Tau Function of 35
Tau Function of 35 0 56613 297742 2017-05-17T20:54:37Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({35}\right) = 4$  where $\tau$ denotes the Definition:Tau Functi..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({35}\right) = 4$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$35 = 5 \times 7$ Thus: :$\tau \left({35}\right) = \tau \left({5^1 \times 7^1}\right) = \left({1 + 1}\right) \left({1 + 1}\right) = 4$ The divisors of $35$ can be enumerated as: :$1, 5, 7, 35$
Tau Function of 120
== Example of Use of $\tau$ Function ==  :$\tau \left({120}\right) = 16$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$120 = 2^3 \times 3 \times 5$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({120}\right)       | r = \tau \left({2^3 \times 3^1 \times 5^1}\right)       | c =  }} {{eqn | r = \left({3 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $120$ can be enumerated as: :$1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$ {{OEIS|A018293}}
Tau Function of 6
== Example of Use of $\tau$ Function ==  :$\tau \left({6}\right) = 4$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$6 = 2 \times 3$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({6}\right)       | r = \tau \left({2^1 \times 3^1}\right)       | c =  }} {{eqn | r = \left({1 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 4       | c =  }} {{end-eqn}} The divisors of $6$ can be enumerated as: :$1, 2, 3, 6$
Tau Function of 28
== Example of Use of $\tau$ Function ==  :$\tau \left({28}\right) = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$28 = 2^2 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({28}\right)       | r = \tau \left({2^2 \times 7^1}\right)       | c =  }} {{eqn | r = \left({2 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $28$ can be enumerated as: :$1, 2, 4, 7, 14, 28$ {{OEIS|A018254}}
Tau Function of 140
== Example of Use of $\tau$ Function ==  :$\tau \left({140}\right) = 12$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$140 = 2^2 \times 5 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({140}\right)       | r = \tau \left({2^2 \times 5^1 \times 7^1}\right)       | c =  }} {{eqn | r = \left({2 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $140$ can be enumerated as: :$1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$ {{OEIS|A018301}}
Tau Function of 270
Tau Function of 270 0 56972 299223 2017-05-26T21:24:58Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({270}\right) = 16$  where $\tau$ denotes the Definition:Tau Func..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({270}\right) = 16$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$270 = 2 \times 3^3 \times 5$ Thus: :$\tau \left({270}\right) = \tau \left({2^1 \times 3^3 \times 5^1}\right) = \left({1 + 1}\right) \left({3 + 1}\right) \left({1 + 1}\right) = 16$ The divisors of $270$ can be enumerated as: :$1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270$
Sigma Function of 270
== Example of Sigma Function of Integer ==  :$\sigma \left({270}\right) = 720$  where $\sigma$ denotes the $\sigma$ function. == Proof == From Sigma Function of Integer :$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$270 = 2 \times 3^3 \times 5$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({270}\right)       | r = \left({2 + 1}\right) \times \frac {3^4 - 1} {3 - 1} \times \left({5 + 1}\right)       | c =  }} {{eqn | r = 3 \times \frac {81 - 1} 2 \times 6       | c =  }} {{eqn | r = 3 \times 40 \times 6       | c =  }} {{eqn | r = 3 \times \left({2^3 \times 5}\right) \times \left({2 \times 3}\right)       | c =  }} {{eqn | r = 2^4 \times 3^2 \times 5       | c =  }} {{eqn | r = 720       | c =  }} {{end-eqn}}
Tau Function of 496
== Example of Use of $\tau$ Function ==  :$\tau \left({496}\right) = 10$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$496 = 2^4 \times 31$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({496}\right)       | r = \tau \left({2^4 \times 31^1}\right)       | c =  }} {{eqn | r = \left({4 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $496$ can be enumerated as: :$1, 2, 4, 8, 16, 31, 62, 124, 248, 496$ {{OEIS|A018487}}
Sigma Function of 496
== Example of Sigma Function of Integer ==  :$\sigma \left({496}\right) = 992$  where $\sigma$ denotes the $\sigma$ function. == Proof == From Sigma Function of Integer :$\displaystyle \sigma \left({n}\right) = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$496 = 2^4 \times 31$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({496}\right)       | r = \left({2^5 - 1}\right) \times \left({31 + 1}\right)       | c =  }} {{eqn | r = 31 \times 32       | c =  }} {{eqn | r = 31 \times 2^5       | c =  }} {{eqn | r = 992       | c =  }} {{end-eqn}}
Euler Phi Function of 164
== Example of Use of Euler $\phi$ Function ==  :$\map \phi {164} = 80$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == From Euler Phi Function of Integer: :$\displaystyle \map \phi n = n \prod_{p \mathop \divides n} \paren {1 - \frac 1 p}$ where $p \divides n$ denotes the primes which divide $n$. We have that: :$164 = 2^2 \times 41$ Thus: {{begin-eqn}} {{eqn | l = \map \phi {164}       | r = 164 \paren {1 - \dfrac 1 2} \paren {1 - \dfrac 1 {41} }       | c =  }} {{eqn | r = 164 \times \frac 1 2 \times \frac {40} {41}       | c =  }} {{eqn | r = 2 \times 1 \times 40       | c =  }} {{eqn | r = 80       | c =  }} {{end-eqn}}
Tau Function of 18
== Example of Use of $\tau$ Function ==  :$\map \tau {18} = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$18 = 2 \times 3^2$ Thus: :$\map \tau {18} = \map \tau {2^1 \times 3^2} = \paren {1 + 1} \paren {2 + 1} = 6$ The divisors of $18$ can be enumerated as: :$1, 2, 3, 6, 9, 18$ {{OEIS|A018251}}
Tau Function of 27
Tau Function of 27 0 57164 299848 2017-06-01T17:38:30Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({27}\right) = 4$  where $\tau$ denotes the Definition:Tau Functi..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({27}\right) = 4$  where $\tau$ denotes the $\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \tau \left({27}\right)       | r = \tau \left({3^3}\right)       | c =  }} {{eqn | r = 3 + 1       | c = Tau of Power of Prime }} {{eqn | r = 4       | c =  }} {{end-eqn}} The divisors of $27$ can be enumerated as: :$1, 3, 9, 27$
Tau Function of 24
== Example of Use of $\tau$ Function ==  :$\map \tau {24} = 8$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$24 = 2^3 \times 3$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {24}       | r = \map \tau {2^3 \times 3^1}       | c =  }} {{eqn | r = \paren {3 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $24$ can be enumerated as: :$1, 2, 3, 4, 6, 8, 12, 24$ {{OEIS|A018253}}
Tau Function of 32
Tau Function of 32 0 57166 299850 2017-06-01T17:45:57Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({32}\right) = 6$  where $\tau$ denotes the Definition:Tau Functi..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({32}\right) = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \tau \left({32}\right)       | r = \tau \left({2^5}\right)       | c =  }} {{eqn | r = 5 + 1       | c = Tau of Power of Prime }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $32$ can be enumerated as: :$1, 2, 4, 8, 16, 32$
Tau Function of 56
== Example of Use of $\tau$ Function ==  :$\map \tau {56} = 8$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$56 = 2^3 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {56}       | r = \map \tau {2^3 \times 7^1}       | c =  }} {{eqn | r = \paren {3 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $56$ can be enumerated as: :$1, 2, 4, 7, 8, 14, 28, 56$ {{OEIS|A018265}}
Tau Function of 64
Tau Function of 64 0 57168 299853 2017-06-01T17:51:50Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({64}\right) = 7$  where $\tau$ denotes the Definition:Tau Functi..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({64}\right) = 7$  where $\tau$ denotes the $\tau$ Function. == Proof == {{begin-eqn}} {{eqn | l = \tau \left({64}\right)       | r = \tau \left({2^6}\right)       | c =  }} {{eqn | r = 6 + 1       | c = Tau of Power of Prime }} {{eqn | r = 7       | c =  }} {{end-eqn}} The divisors of $64$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 64$
Euler Phi Function of 194
== Example of Use of Euler $\phi$ Function ==  :$\phi \left({194}\right) = 96$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == We have that: :$194 = 2 \times 97$ Thus: {{begin-eqn}} {{eqn | l = \phi \left({194}\right)       | r = 97 - 1       | c = Euler Phi Function of 2 times Odd Prime }} {{eqn | r = 96       | c =  }} {{end-eqn}}
Euler Phi Function of 195
== Example of Euler $\phi$ Function of Square-Free Integer ==  :$\map \phi {195} = 96$  where $\phi$ denotes the Euler $\phi$ Function. == Proof == From Euler Phi Function of Square-Free Integer: :$\displaystyle \map \phi n = \prod_{\substack {p \mathop \divides n \\ p \mathop > 2} } \paren {p - 1}$ where $p \divides n$ denotes the primes which divide $n$. We have that: :$195 = 3 \times 5 \times 13$ and so is square-free. Thus: {{begin-eqn}} {{eqn | l = \map \phi {195}       | r = \paren {3 - 1} \paren {5 - 1} \paren {13 - 1}       | c =  }} {{eqn | r = 2 \times 4 \times 12       | c =  }} {{eqn | r = 96       | c =  }} {{end-eqn}}
Tau Function of 70
== Example of Use of $\tau$ Function ==  :$\map \tau {70} = 8$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$70 = 2 \times 5 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {70}       | r = \map \tau {2^1 \times 5^1 \times 7^1}       | c =  }} {{eqn | r = \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $70$ can be enumerated as: :$1, 2, 5, 7, 10, 14, 35, 70$ {{OEIS|A018270}}
Amicable Pair/Examples/220-284
== Example of Amicable Pair ==  $220$ and $284$ are the smallest amicable pair: :$\map \sigma {220} = \map \sigma {284} = 504 = 220 + 284$  == Proof == Let $\map s n$ denote the aliquot sum of $n$. By definition: :$\map s n = \map \sigma n - n$ where $\map \sigma n$ denotes the $\sigma$ function. Thus: {{begin-eqn}} {{eqn | l = \map s {220}       | r = \map \sigma {220} - 220       | c =  }} {{eqn | r = 504 - 220       | c = $\sigma$ of $220$ }} {{eqn | r = 284       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = \map s {284}       | r = \map \sigma {284} - 284       | c =  }} {{eqn | r = 504 - 284       | c = $\sigma$ of $284$ }} {{eqn | r = 220       | c =  }} {{end-eqn}} It can be determined by inspection of the aliquot sums of all smaller integers that there is no smaller amicable pair.  == Historical Note == {{:Amicable Pair/Examples/220-284/Historical Note}} == Sources == * {{BookReference|History of the Theory of Numbers|1919|Leonard Eugene Dickson|volume = I|prev = Definition:Aliquot Part|next = Amicable Pair/Examples/220-284/Historical Note}}: Preface * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 230|next = 220}}: $220$ * {{BookReference|Dictionary of Mathematics|1989|Ephraim J. Borowski|author2 = Jonathan M. Borwein|prev = Definition:Amicable Pair/Definition 1|next = Symbols:Abbreviations/A/amp|entry = amicable numbers}} * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 230|next = 220}}: $220$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Amicable Pair/Historical Note|next = Mathematician:André-Marie Ampère|entry = amicable numbers}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Definition:Amicable Pair/Historical Note|next = Mathematician:André-Marie Ampère|entry = amicable numbers}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Amicable Pair/Definition 1|next = Amicable Pair/Examples/220-284/Historical Note|entry = amicable numbers}} * {{MathWorld|Amicable Pair|AmicablePair}}
Sigma Function of 220
== Example of Sigma Function of Integer ==  :$\map \sigma {220} = 504$  where $\sigma$ denotes the $\sigma$ function. == Proof == From Sigma Function of Integer :$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$220 = 2^2 \times 5 \times 11$ Hence: {{begin-eqn}} {{eqn | l = \map \sigma {220}       | r = \frac {2^3 - 1} {2 - 1} \times \frac {5^2 - 1} {5 - 1} \times \frac {11^2 - 1} {11 - 1}       | c =  }} {{eqn | r = \frac 7 1 \times \frac {6 \times 4} 4 \times \frac {12 \times 10} {10}       | c =  }} {{eqn | r = 7 \times 6 \times 12       | c =  }} {{eqn | r = 7 \times \paren {2 \times 3} \times \paren {2^2 \times 3}       | c =  }} {{eqn | r = 2^3 \times 3^2 \times 7       | c =  }} {{eqn | r = 504       | c =  }} {{end-eqn}}
Sigma Function of 284
== Example of Sigma Function of Integer ==  :$\map \sigma {284} = 504$  where $\sigma$ denotes the $\sigma$ function. == Proof == From Sigma Function of Integer :$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$ where $n = \displaystyle \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i}$ denotes the prime decomposition of $n$. We have that: :$284 = 2^2 \times 71$ Hence: {{begin-eqn}} {{eqn | l = \map \sigma {284}       | r = \frac {2^3 - 1} {2 - 1} \times \paren {71 + 1}       | c =  }} {{eqn | r = 7 \times 72       | c =  }} {{eqn | r = 7 \times \paren {2^3 \times 3^2}       | c =  }} {{eqn | r = 2^3 \times 3^2 \times 7       | c =  }} {{eqn | r = 504       | c =  }} {{end-eqn}}
Tau Function of 242
== Example of Use of $\tau$ Function ==  :$\tau \left({242}\right) = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$242 = 2 \times 11^2$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({242}\right)       | r = \tau \left({2^1 \times 11^2}\right)       | c =  }} {{eqn | r = \left({1 + 1}\right) \left({2 + 1}\right)       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $242$ can be enumerated as: :$1, 2, 11, 22, 121, 242$ {{OEIS|A018351}}
Tau Function of 243
== Example of Use of $\tau$ Function ==  :$\tau \left({243}\right) = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$243 = 3^5$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({243}\right)       | r = \tau \left({3^5}\right)       | c =  }} {{eqn | r = 5 + 1       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $243$ can be enumerated as: :$1, 3, 9, 27, 81, 243$
Tau Function of 244
== Example of Use of $\tau$ Function ==  :$\map \tau {244} = 6$  where $\tau$ denotes the $\tau$ Function. == Proof == From Tau Function from Prime Decomposition: :$\displaystyle \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$ where: :$r$ denotes the number of distinct prime factors in the prime decomposition of $n$ :$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$. We have that: :$244 = 2^2 \times 61$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {244}       | r = \map \tau {2^2 \times 61^1}       | c =  }} {{eqn | r = \paren {2 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 6       | c =  }} {{end-eqn}} The divisors of $244$ can be enumerated as: :$1, 2, 4, 61, 122, 244$ {{OEIS|A018352}}