Primes: Difference between revisions
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<td>returns an | <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>R</apll> is | <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.</td> | <td><apll>Z</apll> is an integer array of the same shape as <apll>R</apll>.</td> | ||
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<p>The Nth 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>The Nth 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> ¯2π1000003<br /> | |||
15485927</apll> | |||
== Previous Prime Function == | == Previous Prime Function == | ||
<p>The previous prime function (<apll>¯1πR</apll>) returns the prime that immediately precedes <apll>R</apll>.</p> | <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> ¯1π1000000<br /> | |||
999983</apll> | |||
== Primality Test == | == 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>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> 0π1000003<br /> | |||
1</apll> | |||
== Next Prime Function == | == Next Prime Function == | ||
<p>The next prime function (<apll>1πR</apll>) returns the prime that immediately follows <apll>R</apll>.</p> | <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> 1π1000000<br /> | |||
1000003</apll> | |||
== Number Of Primes Function == | == 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>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> 2π15485927<br /> | |||
1000003</apll> | |||
== Divisor Count Function == | == Divisor Count Function == | ||
Revision as of 15:55, 8 March 2013
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| R is a scalar or one-element vector consisting of a positive integer to be factored. | ||||
| Z is an integer vector whose values are the prime factors of R. |
For example,
π120
2 2 2 3 5
×/⎕←π⎕←¯1+2*62
4611686018427387903
3 715827883 2147483647
4611686018427387903
π¯1+2*61
2305843009213693951
⍴π1
0
|
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L is an integer scalar whose meaning is as follows
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| R is an array consisting of positive integers to which one of the above functions is applied, element by element. | ||||||||||||||||||
| Z is an integer array of the same shape as R. |
Nth Prime Function
The Nth prime function (¯2πR) returns the Rth prime where 2 is the first prime. This function is sensitive to the index origin.
For example, how many primes are less then or equal to 1000003?
¯2π1000003
15485927
Previous Prime Function
The previous prime function (¯1πR) returns the prime that immediately precedes R.
For example, what is the prime that immediately precedes 1000000?
¯1π1000000
999983
Primality Test
The primality test function (0πR) returns a 1 if R is a prime and 0 if not.
For example, is 1000003 a prime?
0π1000003
1
Next Prime Function
The next prime function (1πR) returns the prime that immediately follows R.
For example, what is the next prime after 1000000?
1π1000000
1000003
Number Of Primes Function
The number of primes function (2πR) returns number of primes less than or equal to R.
For example, what is the 15485927th prime?
2π15485927
1000003
Divisor Count Function
The divisor count function (10πR) returns the number of divisors of a number. It is the same as ×/1+∪⍦πR where πR returns the prime factors of R and ∪⍦ 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 ×/1+ counts them.
Divisor Sum Function
The divisor sum function (11πR) returns the sum of the divisors of a number. It is the same as ×/(¯1+(∪f)*1+∪⍦f)÷¯1+∪f←πR1. This function is used to recognize deficient, perfect, and abundant numbers.
Möbius Function
The Möbius function (12πR) returns information about the square free properties of R. If R is square free, the function returns 1 if R has an even number of prime factors, and ¯1 if it has an odd number of prime factors. If the argument is not square free, the function returns 0. It is used in the Möbius Inversion Formula to invert general arithmetic functions.
Totient Function
The totient function (13πR) (also called Euler's Totient Function) returns the number of positive integers less than or equal to R that are relatively prime to it (i.e., having no common positive factors other than 1).
