hcju/proofwikips huggingface dataset:
Tau Function of 104
== Example of Use of $\tau$ Function ==  :$\tau \left({104}\right) = 8$  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: :$104 = 2^3 \times 13$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({104}\right)       | r = \tau \left({2^3 \times 13^1}\right)       | c =  }} {{eqn | r = \left({3 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 8       | c =  }} {{end-eqn}} The divisors of $104$ can be enumerated as: :$1, 2, 4, 8, 13, 26, 52, 104$ {{OEIS|A018285}}
Sigma Function of 272
== Example of Sigma Function of Integer ==  :$\sigma \left({272}\right) = 558$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$272 = 2^4 \times 17$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({272}\right)       | r = \frac {2^5 - 1} {2 - 1} \times \frac {17^2 - 1} {17 - 1}       | c = Sigma Function of Integer }} {{eqn | r = \frac {31 - 1} 1 \times \frac {289 - 1} {16}       | c =  }} {{eqn | r = 31 \times 18       | c =  }} {{eqn | r = \left({2 \times 3^2}\right) \times 31       | c =  }} {{eqn | r = 558       | c =  }} {{end-eqn}}
Tau Function of 272
Tau Function of 272 0 59027 304242 2017-07-10T09:26:58Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({272}\right) = 10$  where $\tau$ denotes the Definition:Tau Func..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({272}\right) = 10$  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: :$272 = 2^4 \times 17$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({272}\right)       | r = \tau \left({2^4 \times 17^1}\right)       | c =  }} {{eqn | r = \left({4 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $272$ can be enumerated as: :$1, 2, 4, 8, 16, 17, 34, 68, 136, 272$ {{OEIS|A018366}}
Tau Function of 304
== Example of Use of $\tau$ Function ==  :$\map \tau {304} = 10$  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: :$304 = 2^4 \times 19$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {304}       | r = \map \tau {2^4 \times 19^1}       | c =  }} {{eqn | r = \paren {4 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $304$ can be enumerated as: :$1, 2, 4, 8, 16, 19, 38, 76, 152, 304$ {{OEIS|A018383}}
Sigma Function of 304
== Example of Sigma Function of Integer ==  :$\map \sigma {304} = 620$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$304 = 2^4 \times 19$ Hence: {{begin-eqn}} {{eqn | l = \map \sigma {304}       | r = \frac {2^5 - 1} {2 - 1} \times \frac {19^2 - 1} {19 - 1}       | c = Sigma Function of Integer }} {{eqn | r = \frac {31 - 1} 1 \times \frac {361 - 1} {18}       | c =  }} {{eqn | r = 31 \times 20       | c =  }} {{eqn | r = \paren {2^2 \times 5} \times 31       | c =  }} {{eqn | r = 620       | c =  }} {{end-eqn}}
Sigma Function of 748
== Example of Sigma Function of Integer ==  :$\sigma \left({748}\right) = 1512$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$748 = 2^2 \times 11 \times 17$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({748}\right)       | r = \frac {2^3 - 1} {2 - 1} \times \left({11 + 1}\right) \times \left({17 + 1}\right)       | c = Sigma Function of Integer }} {{eqn | r = 7 \times 12 \times 18       | c =  }} {{eqn | r = 7 \times \left({2^2 \times 3}\right) \times \left({2 \times 3^2}\right)       | c =  }} {{eqn | r = 2^3 \times 3^3 \times 7       | c =  }} {{eqn | r = 1512       | c =  }} {{end-eqn}}
Tau Function of 748
== Example of Use of $\tau$ Function ==  :$\tau \left({748}\right) = 12$  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: :$748 = 2^2 \times 11 \times 17$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({748}\right)       | r = \tau \left({2^2 \times 11^1 \times 17^1}\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 $748$ can be enumerated as: :$1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748$ {{OEIS|A018625}}
Perfect Number/Examples/6
== Example of Perfect Number ==  $6$ is a perfect number: :$1 + 2 + 3 = 6$  == Proof == {{begin-eqn}} {{eqn | l = 6       | r = 2 \times 3       | c =  }} {{eqn | r = 2^{2 - 1} \left({2^2 - 1}\right)       | c =  }} {{end-eqn}} Thus $6$ is in the form $2^{n - 1} \left({2^n - 1}\right)$. $\left({2^2 - 1}\right) = 3$ is prime. So $6$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $6$ are enumerated at $\tau$ of $6$: :$1, 2, 3$
Perfect Number/Examples/28
== Example of Perfect Number ==  $28$ is a perfect number: :$1 + 2 + 4 + 7 + 14 = 28$  == Proof == {{begin-eqn}} {{eqn | l = 28       | r = 4 \times 7       | c =  }} {{eqn | r = 2^{3 - 1} \left({2^3 - 1}\right)       | c =  }} {{end-eqn}} Thus $28$ is in the form $2^{p - 1} \left({2^p - 1}\right)$. $\left({2^3 - 1}\right) = 7$ is prime. So $28$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $28$ are enumerated at $\tau$ of $28$: :$1, 2, 4, 7, 14$
Perfect Number/Examples/496
Perfect Number/Examples/496 0 59052 304297 2017-07-10T21:05:03Z Prime.mover 59 Created page with "== Example of Perfect Number ==  $496$ is a perfect number: :$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248..." wikitext text/x-wiki == Example of Perfect Number ==  $496$ is a perfect number: :$1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496$  == Proof == {{begin-eqn}} {{eqn | l = 496       | r = 16 \times 31       | c =  }} {{eqn | r = 2^{5 - 1} \left({2^5 - 1}\right)       | c =  }} {{end-eqn}} Thus $496$ is in the form $2^{p - 1} \left({2^p - 1}\right)$. $\left({2^5 - 1}\right) = 31$ is prime. So $496$ is perfect by the Theorem of Even Perfect Numbers. The aliquot parts of $496$ are enumerated at $\tau$ of $496$: :$1, 2, 4, 8, 16, 31, 62, 124, 248$
Tau Function of 8128
Tau Function of 8128 0 59053 304298 2017-07-10T21:08:43Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({8128}\right) = 10$  where $\tau$ denotes the Definition:Tau Fun..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({8128}\right) = 10$  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: :$8128 = 2^6 \times 127$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({8128}\right)       | r = \tau \left({2^6 \times 127^1}\right)       | c =  }} {{eqn | r = \left({6 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 14       | c =  }} {{end-eqn}} The divisors of $496$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, 8128$ {{OEIS|A133024}}
Tau Function of 350
Tau Function of 350 0 59059 304308 2017-07-10T22:22:33Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({350}\right) = 12$  where $\tau$ denotes the Definition:Tau Func..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({350}\right) = 12$  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: :$350 = 2 \times 5^2 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({350}\right)       | r = \tau \left({2^1 \times 5^2 \times 7^1}\right)       | c =  }} {{eqn | r = \left({1 + 1}\right) \left({2 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $350$ can be enumerated as: :$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350$ {{OEIS|A018406}}
Tau Function of 368
Tau Function of 368 0 59061 304310 2017-07-10T22:28:37Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({368}\right) = 10$  where $\tau$ denotes the Definition:Tau Func..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({368}\right) = 10$  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: :$368 = 2^4 \times 23$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({368}\right)       | r = \tau \left({2^4 \times 23^1}\right)       | c =  }} {{eqn | r = \left({4 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $368$ can be enumerated as: :$1, 2, 4, 8, 16, 23, 46, 92, 184, 368$ {{OEIS|A018416}}
Tau Function of 464
== Example of Use of $\tau$ Function ==  :$\map \tau {464} = 10$  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: :$464 = 2^4 \times 29$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {464}       | r = \map \tau {2^4 \times 29^1}       | c =  }} {{eqn | r = \paren {4 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 10       | c =  }} {{end-eqn}} The divisors of $464$ can be enumerated as: :$1, 2, 4, 8, 16, 29, 58, 116, 232, 464$ {{OEIS|A018469}}
Sigma Function of 464
== Example of Sigma Function of Integer ==  :$\sigma \left({464}\right) = 930$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$464 = 2^4 \times 29$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({464}\right)       | r = \frac {2^5 - 1} {2 - 1} \times \left({29 + 1}\right)       | c = Sigma Function of Integer }} {{eqn | r = \frac {31 - 1} 1 \times 30       | c =  }} {{eqn | r = 31 \times 30       | c =  }} {{eqn | r = \left({2 \times 3 \times 5}\right) \times 31       | c =  }} {{eqn | r = 930       | c =  }} {{end-eqn}}
Sigma Function of 550
== Example of Sigma Function of Integer ==  :$\sigma \left({550}\right) = 1116$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$550 = 2 \times 5^2 \times 11$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({550}\right)       | r = \left({2 + 1}\right) \times \frac {5^3 - 1} {5 - 1} \times \left({11 + 1}\right)       | c = Sigma Function of Integer }} {{eqn | r = 3 \times \frac {125 - 1} 4 \times 12       | c =  }} {{eqn | r = 3 \times 31 \times \left({2^2 \times 3}\right)       | c =  }} {{eqn | r = 2^2 \times 3^2 \times 31       | c =  }} {{eqn | r = 1116       | c =  }} {{end-eqn}}
Sigma Function of 572
== Example of Sigma Function of Integer ==  :$\sigma \left({572}\right) = 1176$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$572 = 2^2 \times 11 \times 13$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({572}\right)       | r = \frac {2^3 - 1} {2 - 1} \times \left({11 + 1}\right) \times \left({13 + 1}\right)       | c = Sigma Function of Integer }} {{eqn | r = \frac {7 - 1} 1 \times 12 \times 14       | c =  }} {{eqn | r = 7 \times \left({2^2 \times 3}\right) \times \left({2 \times 7}\right)       | c =  }} {{eqn | r = 2^3 \times 3 \times 7^2       | c =  }} {{eqn | r = 1176       | c =  }} {{end-eqn}}
Sigma Function of 650
== Example of Sigma Function of Integer ==  :$\map \sigma {650} = 1302$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$650 = 2 \times 5^2 \times 13$ Hence: {{begin-eqn}} {{eqn | l = \map \sigma {650}       | r = \paren {2 + 1} \times \frac {5^3 - 1} {5 - 1} \times \paren {13 + 1}       | c = Sigma Function of Integer }} {{eqn | r = 3 \times \frac {125 - 1} 4 \times 14       | c =  }} {{eqn | r = 3 \times 31 \times \paren {2 \times 7}       | c =  }} {{eqn | r = 2 \times 3 \times 7 \times 31       | c =  }} {{eqn | r = 1302       | c =  }} {{end-eqn}}
Tau Function of 770
== Example of Use of $\tau$ Function ==  :$\tau \left({770}\right) = 16$  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: :$770 = 2 \times 5 \times 7 \times 11$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({770}\right)       | r = \tau \left({2^1 \times 5^1 \times 7^1 \times 11^1}\right)       | c =  }} {{eqn | r = \left({1 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right)       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $770$ can be enumerated as: :$1, 2, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770$ {{OEIS|A018636}}
Multiplicative Persistence/Examples/25
== Examples of Multiplicative Persistence ==  $25$ is the smallest positive integer which has a multiplicative persistence of $2$.  == Proof == Trivially: {{begin-eqn}} {{eqn | n = 1       | l = 2 \times 5       | r = 10 }} {{eqn | n = 2       | l = 1 \times 0       | r = 0 }} {{end-eqn}} All positive integers between $10$ and $19$ are seen to have a multiplicative persistence of $1$: :$1 \times n = n$ where $n$ is a single digit. Then for 2-digit positive integers starting with $2$: :$2 \times n > 9 \implies n > 4$ by inspection. Hence the result.
De Polignac's Formula/Examples/2 in 720 Factorial
== Example of Use of De Polignac's Formula ==  The prime factor $2$ appears in $720!$ to the power of $716$. That is: :$2^{716} \divides 720!$ but: :$2^{717} \nmid 720!$  == Proof == Let $\mu$ denote the power of $2$ which divides $720!$ {{begin-eqn}} {{eqn | l = \mu       | r = \sum_{k \mathop > 0} \floor {\frac {720} {2^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {720} 2} + \floor {\frac {720} 4} + \floor {\frac {720} 8} + \floor {\frac {720} {16} } + \floor {\frac {720} {32} }       | c =  }} {{eqn | o =        | r = + \floor {\frac {720} {64} } + \floor {\frac {720} {128} } + \floor {\frac {720} {256} } + \floor {\frac {720} {512} }       | c =  }} {{eqn | r = 360 + 180 + 90 + 45 + 22 + 11 + 5 + 2 + 1       | c =  }} {{eqn | r = 716       | c =  }} {{end-eqn}}
De Polignac's Formula/Examples/3 in 720 Factorial
== Example of Use of De Polignac's Formula ==  The prime factor $3$ appears in $720!$ to the power of $356$. That is: :$3^{356} \divides 720!$ but: :$3^{357} \nmid 720!$  == Proof == Let $\mu$ denote the power of $3$ which divides $720!$ {{begin-eqn}} {{eqn | l = \mu       | r = \sum_{k \mathop > 0} \floor {\frac {720} {3^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {720} 3} + \floor {\frac {720} 9} + \floor {\frac {720} {27} } + \floor {\frac {720} {81} } + \floor {\frac {720} {243} }       | c =  }} {{eqn | r = 240 + 80 + 26 + 8 + 2       | c =  }} {{eqn | r = 356       | c =  }} {{end-eqn}}
De Polignac's Formula/Examples/5 in 720 Factorial
== Example of Use of De Polignac's Formula ==  The prime factor $5$ appears in $720!$ to the power of $178$. That is: :$5^{178} \divides 720!$ but: :$5^{179} \nmid 720!$  == Proof == Let $\mu$ denote the power of $5$ which divides $720!$ {{begin-eqn}} {{eqn | l = \mu       | r = \sum_{k \mathop > 0} \floor {\frac {720} {5^k} }       | c = De Polignac's Formula }} {{eqn | r = \floor {\frac {720} 5} + \floor {\frac {720} {25} } + \floor {\frac {720} {125} } + \floor {\frac {720} {625} }       | c =  }} {{eqn | r = 144 + 28 + 5 + 1       | c =  }} {{eqn | r = 178       | c =  }} {{end-eqn}}
Sigma Function of 1184
== Example of Sigma Function of Integer ==  :$\map \sigma {1184} = 2394$  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: :$1184 = 2^5 \times 37$ Hence: {{begin-eqn}} {{eqn | l = \map \sigma {1184}       | r = \frac {2^6 - 1} {2 - 1} \times \frac {37^2 - 1} {37 - 1}       | c =  }} {{eqn | r = \frac {63} 1 \times \frac {38 \times 36} {36}       | c =  }} {{eqn | r = 63 \times 38       | c =  }} {{eqn | r = \paren {3^2 \times 7} \times \paren {2 \times 19}       | c =  }} {{eqn | r = 2 \times 3^2 \times 7 \times 19       | c =  }} {{eqn | r = 2394       | c =  }} {{end-eqn}}
Sigma Function of 1210
== Example of Sigma Function of Integer ==  :$\sigma \left({1210}\right) = 2394$  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: :$1210 = 2 \times 5 \times 11^2$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({1210}\right)       | r = \left({2 + 1}\right) \times \left({5 + 1}\right) \times \frac {11^3 - 1} {11 - 1}       | c =  }} {{eqn | r = 3 \times 6 \times \frac {1330} {10}       | c =  }} {{eqn | r = 3 \times 6 \times 133       | c =  }} {{eqn | r = 3 \times \left({2 \times 3}\right) \times \left({7 \times 19}\right)       | c =  }} {{eqn | r = 2 \times 3^2 \times 7 \times 19       | c =  }} {{eqn | r = 2394       | c =  }} {{end-eqn}}
Sigma Function of 1638
== Example of Sigma Function of Integer ==  :$\sigma \left({1638}\right) = 4368$  where $\sigma$ denotes the $\sigma$ function. == Proof == We have that: :$1638 = 2 \times 3^2 \times 7 \times 13$ Hence: {{begin-eqn}} {{eqn | l = \sigma \left({1638}\right)       | r = \left({2 + 1}\right) \frac {3^3 - 1} {3 - 1} \times \left({7 + 1}\right) \times \left({13 + 1}\right)       | c = Sigma Function of Integer }} {{eqn | r = 3 \times \frac {26} 2 \times 8 \times 14       | c =  }} {{eqn | r = 3 \times 13 \times 2^3 \times \left({2 \times 7}\right)       | c =  }} {{eqn | r = 2^4 \times 3 \times 7 \times 13       | c =  }} {{eqn | r = 4368       | c =  }} {{end-eqn}}
Tau Function of 1638
== Example of Use of $\tau$ Function ==  :$\map \tau {1638} = 24$  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: :$1638 = 2 \times 3^2 \times 7 \times 13$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {1638}       | r = \map \tau {2^1 \times 3^2 \times 7^1 \times 13^1}       | c =  }} {{eqn | r = \paren {1 + 1} \paren {2 + 1} \paren {1 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 24       | c =  }} {{end-eqn}} The divisors of $1638$ can be enumerated as: :$1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638$
Tau Function of 945
== Example of Use of $\tau$ Function ==  :$\map \tau {945} = 16$  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: :$945 = 3^3 \times 5 \times 7$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {945}       | r = \map \tau {3^3 \times 5^1 \times 7^1}       | c =  }} {{eqn | r = \paren {3 + 1} \paren {1 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $945$ can be enumerated as: :$1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945$ {{OEIS|A018736}}
Tau Function of 910
== Example of Use of $\tau$ Function ==  :$\map \tau {910} = 16$  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: :$910 = 2 \times 5 \times 7 \times 13$ Thus: {{begin-eqn}} {{eqn | l = \map \tau {910}       | r = \map \tau {2^1 \times 5^1 \times 7^1 \times 13^1}       | c =  }} {{eqn | r = \paren {1 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}       | c =  }} {{eqn | r = 16       | c =  }} {{end-eqn}} The divisors of $910$ can be enumerated as: :$1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910$ {{OEIS|A018717}}
Tau Function of 1184
Tau Function of 1184 0 60611 310282 2017-08-10T06:33:21Z Prime.mover 59 Created page with "== Example of Use of $\tau$ Function ==  :$\tau \left({1184}\right) = 12$  where $\tau$ denotes the Definition:Tau Fun..." wikitext text/x-wiki == Example of Use of $\tau$ Function ==  :$\tau \left({1184}\right) = 12$  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: :$1184 = 2^5 \times 37$ Thus: {{begin-eqn}} {{eqn | l = \tau \left({1184}\right)       | r = \tau \left({2^5 \times 37^1}\right)       | c =  }} {{eqn | r = \left({5 + 1}\right) \left({2 + 1}\right)       | c =  }} {{eqn | r = 12       | c =  }} {{end-eqn}} The divisors of $1184$ can be enumerated as: :$1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184$
Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999/Examples
== Examples of Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999 ==  {{begin-eqn}} {{eqn | l = 4 \times 999       | r = 3996       | c =  }} {{eqn | ll= \leadsto       | l = 3 + 996       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 15 \times 999       | r = 14 \, 985       | c =  }} {{eqn | ll= \leadsto       | l = 14 + 985       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 47 \times 999       | r = 46 \, 953       | c =  }} {{eqn | ll= \leadsto       | l = 46 + 953       | r = 999       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 57 \times 999       | r = 56 \, 943       | c =  }} {{eqn | ll= \leadsto       | l = 56 + 943       | r = 999       | c =  }} {{end-eqn}}