View source for Primes

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      <td valign="top"><apll>Z←πR</apll></td>
      <td></td>
      <td></td>
      <td>returns an integer vector consisting of the prime factors of <apll>R</apll>.</td>
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  <td><apll>R</apll> is a scalar or one-element vector consisting of a positive integer to be factored.</td>
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  <td><apll>Z</apll> is an integer vector whose values are the prime factors of <apll>R</apll>.</td>
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<p>For example,</p>

<apll><pre>
      π120
2 2 2 3 5

      ×/⎕←π⎕←¯1+2*62
4611686018427387903
3 715827883 2147483647
4611686018427387903

      π¯1+2*61
2305843009213693951

      ⍴π1
0</pre></apll>




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      <td valign="top"><apll>Z←LπR</apll></td>
      <td></td>
      <td></td>
      <td>returns an array whose values depend upon which number-theoretic function chosen by <apll>L</apll> is applied to <apll>R</apll>.</td>
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  <td><apll>L</apll> is an integer scalar whose meaning is as follows
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        <td><apll>¯2</apll></td>
        <td>&nbsp;&nbsp;Rth prime function</td>
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        <td><apll>¯1</apll></td>
        <td>&nbsp;&nbsp;Previous prime function</td>
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        <td><apll>&nbsp;0</apll></td>
        <td>&nbsp;&nbsp;Primality test function</td>
      </tr>

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        <td><apll>&nbsp;1</apll></td>
        <td>&nbsp;&nbsp;Next prime function</td>
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        <td><apll>&nbsp;2</apll></td>
        <td>&nbsp;&nbsp;Number of primes function</td>
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        <td><apll>10</apll></td>
        <td>&nbsp;&nbsp;Divisor count function</td>
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        <td><apll>11</apll></td>
        <td>&nbsp;&nbsp;Divisor sum function</td>
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        <td><apll>12</apll></td>
        <td>&nbsp;&nbsp;Möbius function function</td>
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        <td><apll>13</apll></td>
        <td>&nbsp;&nbsp;Euler totient function</td>
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  <td><apll>R</apll> is an array consisting of positive integers to which one of the above functions is applied, element by element.</td>
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  <td><apll>Z</apll> is an integer array of the same shape as <apll>R</apll>.</td>
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== Rth Prime Function ==

<p>The Rth prime function (<apll>¯2πR</apll>) returns the <apll>R</apll><sup>th</sup> prime where <apll>2</apll> is the first prime.  This function is sensitive to the index origin.</p>

<p>For example, how many primes are less then or equal to <apll>1000003</apll>?</p>

<apll><pre>
      ¯2π1000003
15485927</pre></apll>

== Previous Prime Function ==

<p>The previous prime function (<apll>¯1πR</apll>) returns the prime that immediately precedes <apll>R</apll>.</p>

<p>For example, what is the prime that immediately precedes <apll>1000000</apll>?</p>

<apll><pre>
      ¯1π1000000
999983</pre></apll>

== Primality Test ==

<p>The primality test function (<apll>0πR</apll>) returns a <apll>1</apll> if <apll>R</apll> is a prime and <apll>0</apll> if not.</p>

<p>For example, is <apll>1000003</apll> a prime?</p>

<apll><pre>
      0π1000003
1</pre></apll>

<p>List the primes up to 100</p>

<apll><pre>
      ⍸0π⍳100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97</pre></apll>

== Next Prime Function ==

<p>The next prime function (<apll>1πR</apll>) returns the prime that immediately follows <apll>R</apll>.</p>

<p>For example, what is the next prime after <apll>1000000</apll>?</p>

<apll><pre>
      1π1000000
1000003</pre></apll>

== Number Of Primes Function ==

<p>The number of primes function (<apll>2πR</apll>) returns number of primes less than or equal to <apll>R</apll>.</p>

<p>For example, what is the <apll>15485927</apll>th prime?</p>

<apll><pre>
      2π15485927
1000003</pre></apll>

== Divisor Count Function ==

<p>The divisor count function (<apll>10πR</apll>) returns the number of divisors of a number.  It is the same as <apll>×/1+∪⍦πR</apll> where <apll>πR</apll> returns the prime factors of <apll>R</apll> and <apll>∪⍦</apll> counts the number of occurrences of unique elements (in this case, the exponent vector of the unique primes).  A divisor then consists of the product of zero or more of the unique primes which is why <apll>×/1+</apll> counts them.</p>

== Divisor Sum Function ==

<p>The divisor sum function (<apll>11πR</apll>) returns the sum of the divisors of a number.  It is the same as <apll>×/(¯1+(∪f)*1+∪⍦f)÷¯1+∪f←πR</apll><sup>1</sup>.  This function is used to recognize [http://en.wikipedia.org/wiki/Deficient_number deficient], [http://en.wikipedia.org/wiki/Perfect_number perfect], and [http://en.wikipedia.org/wiki/Abundant_number abundant] numbers.</p>

== Möbius Function ==

<p>The [http://en.wikipedia.org/wiki/M%C3%B6bius_function Möbius function] (<apll>12πR</apll>) returns information about the square free properties of <apll>R</apll>.  If <apll>R</apll> is [http://en.wikipedia.org/wiki/Square-free_integer square free], the function returns <apll>1</apll> if <apll>R</apll> has an even number of prime factors, and <apll>¯1</apll> if it has an odd number of prime factors.  If the argument is not square free, the function returns <apll>0</apll>.  It is used in the [http://en.wikipedia.org/wiki/M%C3%B6bius_inversion_formula Möbius Inversion Formula] to invert general arithmetic functions.</p>

== Totient Function ==

<p>The totient function (<apll>13πR</apll>) (also called [http://en.wikipedia.org/wiki/Euler_phi Euler's Totient Function]) returns the number of positive integers less than or equal to <apll>R</apll> that are relatively prime to it (i.e., having no common positive factors other than 1).
</p>

== Examples ==

<p>Add together the first 100 primes:</p>

<p>The Rth prime function (<apll>¯2πR</apll>) gives the value of the Rth prime, as in</p>

<apll><pre>
      ¯2π100
541</pre></apll>

<p>The Index function (<apll>⍳R</apll>) produces a vector of integers of length <apll>R</apll>, as in</p>

<apll><pre>
      ⍳¯2π100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... 541</pre></apll>

<p>The Primality Test function (<apll>0πR</apll>) returns a <apll>1</apll> if the corresponding element in <apll>R</apll> is a prime, <apll>0</apll> otherwise, as in</p>

<apll><pre>
      0π⍳¯2π100
0 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 ... 1</pre></apll>

<p>The Indices function (<apll>⍸R</apll>) converts the argument <apll>R</apll> to indices (equivalent to <apll>(,R)/⍳×/⍴R</apll>), as in</p>

<apll><pre>
      ⍸0π⍳¯2π100
2 3 5 7 11 13 17 19 23 29 31 37 41 43 ... 541</pre></apll>

<p>Finally, those numbers may be added together using plus reduction (<apll>+/</apll>), as in</p>

<apll><pre>
      +/⍸0π⍳¯2π100
24133</pre></apll>

== Notes ==

<sup>1</sup> [https://en.wikipedia.org/wiki/Divisor_function Formula for sum of divisors]