Template:Prime factors (with multiplicity)

The following documentation is located at Template:Prime factors (with multiplicity)/doc. [<Edit> Template:Prime factors (with multiplicity)/doc]
[⧼Purge⧽ Template:Prime factors (with multiplicity)]

The {{prime factors (with multiplicity)}} (or {{mpf}} or {{factorization}}) arithmetic function template returns the prime factorization (prime factors (with multiplicity)) of a nonzero integer, otherwise returns an error message.

Usage

{{prime factors (with multiplicity)|a nonzero integer|sep = map items separator (default &#32;*&#32;)|key/val_sep = key/value separator (default ^)}}

or

{{prime factors (with multiplicity)|a nonzero integer|map items separator (default &#32;*&#32;)|key/value separator (default ^)}}

or

{{mpf|a nonzero integer|sep = map items separator (default &#32;*&#32;)|key/val_sep = key/value separator (default ^)}}

or

{{mpf|a nonzero integer|map items separator (default &#32;*&#32;)|key/value separator (default ^)}}

where

• sep default &#32;*&#32; gives " * " (asterisk surrounded by spaces), and
• key/val_sep default is ^.

Valid input

A nonzero integer less than 1031 2 = 1062961.

Examples

To check the results, you may use

• https://oeis.org/A000040/a000040.txt N. J. A. Sloane, Table of n, prime(n) for n = 1..100000
• Andrew Hodges, Java Applet for Factorization
• http://factordb.com/

Examples with valid input

Note that the prime factorization of integer units −1 and 1 gives the empty product of primes, hence the empty string is returned.

Code	Result
{{factorization|210^2}}	2^2 * 3^2 * 5^2 * 7^2
{{mpf|210^2}}	2^2 * 3^2 * 5^2 * 7^2
{{mpf|210^2|sep = ; |key/val_sep =  → }}	2 → 2; 3 → 2; 5 → 2; 7 → 2
{{mpf|210^2|sep = , |key/val_sep = : }}	2: 2, 3: 2, 5: 2, 7: 2
{{mpf|210^2|sep =  * |key/val_sep = ^}}	2^2 * 3^2 * 5^2 * 7^2
{{mpf|210^2|sep =  + |key/val_sep = *}}	2*2 + 3*2 + 5*2 + 7*2
{{mpf|-28}}	2^2 * 7^1
{{mpf|-5}}	5^1
{{mpf|1}}
{{mpf|7}}	7^1
{{mpf|15}}	3^1 * 5^1
{{mpf|27}}	3^3
{{mpf|30}}	2^1 * 3^1 * 5^1
{{mpf|111}}	3^1 * 37^1
{{mpf|5^3 * 11^2}}	5^3 * 11^2
{{mpf|2^5 * 3^3 * 5}}	2^5 * 3^3 * 5^1
{{mpf|2^9 * 3^3}}	2^9 * 3^3
{{mpf|37^2 + 8 * 37^2}}	3^2 * 37^2
{{mpf|2^9 * (26 + 1)}}	2^9 * 3^3
{{mpf|89 * 113}}	89^1 * 113^1
{{mpf|79 * 79}}	79^2
{{mpf|210^2}}	2^2 * 3^2 * 5^2 * 7^2
{{mpf|233^2}}	233^2
{{mpf|10000}}	2^4 * 5^4
{{mpf|65535}}	3^1 * 5^1 * 17^1 * 257^1
{{mpf|65536}}	2^16
{{mpf|65537}}	65537^1
{{mpf|65539}}	65539^1
{{mpf|65540}}	2^2 * 5^1 * 29^1 * 113^1
{{mpf|65541}}	3^1 * 7^1 * 3121^1
{{mpf|65542}}	2^1 * 32771^1
{{mpf|65543}}	65543^1
{{mpf|65547}}	3^2 * 7283^1
{{mpf|65549}}	11^1 * 59^1 * 101^1
{{mpf|65551}}	65551^1
{{mpf|65553}}	3^1 * 21851^1
{{mpf|65557}}	65557^1
{{mpf|65559}}	3^1 * 13^1 * 41^2
{{mpf|65561}}	53^1 * 1237^1
{{mpf|65563}}	65563^1
{{mpf|65567}}	173^1 * 379^1
{{mpf|65569}}	7^1 * 17^1 * 19^1 * 29^1
{{mpf|65571}}	3^1 * 11^1 * 1987^1
{{mpf|65573}}	23^1 * 2851^1
{{mpf|65577}}	3^1 * 21859^1
{{mpf|65579}}	65579^1
{{mpf|265535}}	5^1 * 23^1 * 2309^1
{{mpf|265536}}	2^6 * 3^2 * 461^1
{{mpf|265537}}	131^1 * 2027^1
{{mpf|265539}}	3^1 * 88513^1
{{mpf|265540}}	2^2 * 5^1 * 11^1 * 17^1 * 71^1
{{mpf|265541}}	265541^1
{{mpf|265542}}	2^1 * 3^1 * 44257^1
{{mpf|265543}}	265543^1
{{mpf|265547}}	265547^1
{{mpf|265549}}	37^1 * 7177^1
{{mpf|265551}}	3^1 * 11^1 * 13^1 * 619^1
{{mpf|265553}}	29^1 * 9157^1
{{mpf|265557}}	3^1 * 17^1 * 41^1 * 127^1
{{mpf|265559}}	7^1 * 59^1 * 643^1
{{mpf|265561}}	265561^1
{{mpf|265563}}	3^2 * 19^1 * 1553^1
{{mpf|265567}}	265567^1
{{mpf|265569}}	3^1 * 88523^1
{{mpf|265571}}	265571^1
{{mpf|265573}}	7^1 * 11^1 * 3449^1
{{mpf|265577}}	13^1 * 31^1 * 659^1
{{mpf|265579}}	265579^1
{{mpf|257}}	257^1
{{mpf|97 * 211}}	97^1 * 211^1
{{mpf|216 * 211}}	2^3 * 3^3 * 211^1
{{mpf|1024 * 45}}	2^10 * 3^2 * 5^1
{{mpf|97 * 257}}	97^1 * 257^1
{{mpf|3^6 * 5^2}}	3^6 * 5^2
{{mpf|3 * 5^5}}	3^1 * 5^5
{{mpf|17^2 * 191}}	17^2 * 191^1
{{mpf|5 * 7 * 13 * 29}}	5^1 * 7^1 * 13^1 * 29^1
{{mpf|509^2}}	509^2
{{mpf|965535}}	3^1 * 5^1 * 59^1 * 1091^1
{{mpf|965536}}	2^5 * 11^1 * 13^1 * 211^1
{{mpf|965537}}	67^1 * 14411^1
{{mpf|965539}}	83^1 * 11633^1
{{mpf|965540}}	2^2 * 5^1 * 23^1 * 2099^1
{{mpf|965541}}	3^1 * 321847^1
{{mpf|965542}}	2^1 * 19^1 * 25409^1
{{mpf|965543}}	383^1 * 2521^1
{{mpf|965547}}	3^3 * 11^1 * 3251^1
{{mpf|965549}}	13^1 * 17^2 * 257^1
{{mpf|965551}}	965551^1
{{mpf|965553}}	3^1 * 321851^1
{{mpf|965557}}	31^1 * 31147^1
{{mpf|965559}}	3^1 * 7^1 * 45979^1
{{mpf|965561}}	19^1 * 89^1 * 571^1
{{mpf|965563}}	23^1 * 41981^1
{{mpf|965567}}	965567^1
{{mpf|965569}}	11^1 * 61^1 * 1439^1
{{mpf|965571}}	3^1 * 73^1 * 4409^1
{{mpf|965573}}	7^1 * 271^1 * 509^1
{{mpf|965577}}	3^1 * 367^1 * 877^1
{{mpf|965579}}	193^1 * 5003^1
{{mpf|997 * 1019}}	997^1 * 1019^1
{{mpf|1015943}}	997^1 * 1019^1

Examples with invalid input

Code	Result	Comments
{{mpf|0}}		(undefined prime factorization)
{{mpf|1031^2}}

Formatted numbers

This template requires unformatted numbers, it will not recognize formatted numbers, e.g. comma separated, which is by design since formatted numbers will break expression parsers. To remove the formatting from a number, you can wrap the number first in {{formatnum:number|R}}.

code	result
{{prime factors (with multiplicity)|1,000}}
{{prime factors (with multiplicity)|{{formatnum:1,000|R}}}}	2^3 * 5^3

Code

{{Prime factors (with multiplicity)}} (or {{mpf}} or {{factorization}}) arithmetic function template

{{ifint| {{{1|NAN}}}
| {{#ifexpr: ( ( abs ({{{1}}}) ) > 1 ) and ( ( abs ({{{1}}}) ) < 1031^2 ) 
  | {{~mpf
    | {{{1}}}
    | mpf_le_sqrt(n) = {{mpf le sqrt(n)
                       | {{{1}}}
                       | sep = *
                       | key/val_sep = ^
                       }}
    | sep = {{{sep|{{{2| * }}}}}}
    | key/val_sep = {{{key/val_sep|{{{3|^}}}}}}
    }}
  | {{#ifexpr: ({{{1}}}) = 0
    | {{error| Prime factors (with multiplicity) error: Argument must be a nonzero integer }}
    }}<!--
 -->{{#ifexpr: abs ({{{1}}}) >= 1031^2
    | {{error| Prime factors (with multiplicity) error: Argument must be a nonzero integer with absolute value < 1031{{^|2}} {{=}} {{#expr: 1031^2 }} }}
    }}
  }} 
| {{error| Prime factors (with multiplicity) error: Argument must be a nonzero integer }}
}}

{{~Prime factors (with multiplicity)}} (or {{~mpf}}) core function template

{{#replace:{{#replace:{{{mpf_le_sqrt(n)}}}|*|{{{sep| * }}}}}|^|{{{key/val_sep|^}}}}}<!--
-->{{#if: {{{mpf_le_sqrt(n)}}}
| {{#ifexpr: ( ( abs ({{{1}}}) ) / ( {{{mpf_le_sqrt(n)}}} ) ) > 1
  | {{{sep| * }}}{{#expr: ( abs ({{{1}}}) ) / ( {{{mpf_le_sqrt(n)}}} ) }}{{{key/val_sep|^}}}1
  | <!-- no prime factor greater than sqrt(n) needs to be added -->
  }}
| {{#ifexpr: ( abs ({{{1}}}) ) > 1
  | {{#expr: abs ({{{1}}}) }}{{{key/val_sep|^}}}1 <!-- A prime number! --> 
  | <!-- empty list: empty product of primes for the units (+/-)1, undefined prime factorization for 0 -->
  }}  
}}

See also

• {{if prime}} or {{is prime}}

• {{distinct prime factors up to sqrt(n)}} or {{dpf le sqrt(n)}}
• {{distinct nontrivial prime factors}} or {{dpf lt n}}
• {{distinct prime factors}} or {{dpf}}
• {{number of distinct prime factors}} or {{little omega}}
• {{sum of distinct prime factors}} or {{sodpf}}
• {{product of distinct prime factors}} or {{squarefree kernel}} or {{radical}} or {{rad}}

• {{multiplicity}}

• {{prime factors (with multiplicity) up to sqrt(n)}} or {{mpf le sqrt(n)}}
• {{nontrivial prime factors (with multiplicity)}} or {{mpf lt n}}
• {{prime factors (with multiplicity)}} or {{mpf}} or {{factorization}}
• {{number of prime factors (with multiplicity)}} or {{big Omega}}
• {{sum of prime factors (with multiplicity)}} or {{sopfr}} or {{integer log}}
• {{product of prime factors (with multiplicity)}} (must give back {{abs|n}}, the absolute value of

  n

  )

• {{quadratfrei}}
• {{Moebius mu}} or {{mu}}

• {{Euler phi}} or {{totient}}
• {{Dedekind psi}}

• {{number of divisors}} or {{sigma 0}} or {{tau}}
• {{sum of divisors}} or {{sigma 1}} or {{sigma}} (Cf. {{divisor function}} or {{sigma k}}, with

  k = 1

  (default value))
• {{divisor function}} or {{sigma k}} (for

  k   ≠   0

  )

External links

• Andrew Hodges, Java Applet for Factorization
• http://factordb.com/