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Template:Prime factors (with multiplicity) up to sqrt(n)
From OeisWiki
The following documentation is located at Template:Prime factors (with multiplicity) up to sqrt(n)/doc. [<Edit> Template:Prime factors (with multiplicity) up to sqrt(n)/doc]
[⧼Purge⧽ Template:Prime factors (with multiplicity) up to sqrt(n)]
[⧼Purge⧽ Template:Prime factors (with multiplicity) up to sqrt(n)]
The {{prime factors (with multiplicity) up to sqrt(n)}} (or {{mpf le sqrt(n)}}) arithmetic function template returns a map of prime factors (with multiplicity) up to sqrt(n) of a nonzero integer, otherwise returns an error message.
Contents
Usage
{{prime factors (with multiplicity) up to sqrt(n)|a nonzero integer|sep = map items separator (default , )|key/val_sep = key/value separator (default : ) }}
or
{{prime factors (with multiplicity) up to sqrt(n)|a nonzero integer|map items separator (default , )|key/value separator (default : ) }}
or
{{mpf le sqrt(n)|a nonzero integer|sep = map items separator (default , )|key/val_sep = key/value separator (default : ) }}
or
{{mpf le sqrt(n)|a nonzero integer|map items separator (default , )|key/value separator (default : ) }}
where
- sep default ,  gives , (comma followed by space), and
- key/val_sep default :  gives : (colon followed by space).
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
Examples with valid input (the nonzero noncomposite numbers (units and primes) have empty maps)
Code Result {{mpf le sqrt(n)|-28}} 2: 2 {{mpf le sqrt(n)|-5}} {{mpf le sqrt(n)|1}} {{mpf le sqrt(n)|7}} {{mpf le sqrt(n)|15}} 3: 1 {{mpf le sqrt(n)|27}} 3: 3 {{mpf le sqrt(n)|30}} 2: 1, 3: 1, 5: 1 {{mpf le sqrt(n)|97}} {{mpf le sqrt(n)|111}} 3: 1 {{mpf le sqrt(n)|199}} {{mpf le sqrt(n)|211}} {{mpf le sqrt(n)|5^3 * 11^2}} 5: 3, 11: 2 {{mpf le sqrt(n)|2^5 * 3^3 * 5}} 2: 5, 3: 3, 5: 1 {{mpf le sqrt(n)|2^5 * 3^3 * 5|; | → }} 2 → 5; 3 → 3; 5 → 1 {{mpf le sqrt(n)|2^5 * 3^3 * 5|sep =  * |key/val_sep = ^}} 2^5 * 3^3 * 5^1 {{mpf le sqrt(n)|2^5 * 3^3 * 5|sep =  + |key/val_sep = *}} 2*5 + 3*3 + 5*1 {{mpf le sqrt(n)|2^9 * 3^3}} 2: 9, 3: 3 {{mpf le sqrt(n)|37^2 + 8 * 37^2}} 3: 2, 37: 2 {{mpf le sqrt(n)|2^9 * (26 + 1)}} 2: 9, 3: 3 {{mpf le sqrt(n)|89 * 113}} 89: 1 {{mpf le sqrt(n)|79 * 79}} 79: 2 {{mpf le sqrt(n)|210^2}} 2: 2, 3: 2, 5: 2, 7: 2 {{mpf le sqrt(n)|233^2}} 233: 2 {{mpf le sqrt(n)|10000}} 2: 4, 5: 4 {{mpf le sqrt(n)|65536}} 2: 16 {{mpf le sqrt(n)|65537}} {{mpf le sqrt(n)|65539}} {{mpf le sqrt(n)|65541}} 3: 1, 7: 1 {{mpf le sqrt(n)|65543}} {{mpf le sqrt(n)|65547}} 3: 2 {{mpf le sqrt(n)|65549}} 11: 1, 59: 1, 101: 1 {{mpf le sqrt(n)|65551}} {{mpf le sqrt(n)|65553}} 3: 1 {{mpf le sqrt(n)|65557}} {{mpf le sqrt(n)|65559}} 3: 1, 13: 1, 41: 2 {{mpf le sqrt(n)|65561}} 53: 1 {{mpf le sqrt(n)|65563}} {{mpf le sqrt(n)|65567}} 173: 1 {{mpf le sqrt(n)|65569}} 7: 1, 17: 1, 19: 1, 29: 1 {{mpf le sqrt(n)|265536}} 2: 6, 3: 2, 461: 1 {{mpf le sqrt(n)|265537}} 131: 1 {{mpf le sqrt(n)|265539}} 3: 1 {{mpf le sqrt(n)|265541}} {{mpf le sqrt(n)|265543}} {{mpf le sqrt(n)|265547}} {{mpf le sqrt(n)|265549}} 37: 1 {{mpf le sqrt(n)|265551}} 3: 1, 11: 1, 13: 1 {{mpf le sqrt(n)|265553}} 29: 1 {{mpf le sqrt(n)|265557}} 3: 1, 17: 1, 41: 1, 127: 1 {{mpf le sqrt(n)|265559}} 7: 1, 59: 1 {{mpf le sqrt(n)|265561}} {{mpf le sqrt(n)|265563}} 3: 2, 19: 1 {{mpf le sqrt(n)|265567}} {{mpf le sqrt(n)|265569}} 3: 1 {{mpf le sqrt(n)|97 * 211}} 97: 1 {{mpf le sqrt(n)|216 * 211}} 2: 3, 3: 3, 211: 1 {{mpf le sqrt(n)|1024 * 45}} 2: 10, 3: 2, 5: 1 {{mpf le sqrt(n)|97 * 257}} 97: 1 {{mpf le sqrt(n)|3^6 * 5^2}} 3: 6, 5: 2 {{mpf le sqrt(n)|3 * 5^5}} 3: 1, 5: 5 {{mpf le sqrt(n)|17^2 * 191}} 17: 2, 191: 1 {{mpf le sqrt(n)|5 * 7 * 13 * 29}} 5: 1, 7: 1, 13: 1, 29: 1 {{mpf le sqrt(n)|5 * 7 * 13 * 29|sep = ; }} 5: 1; 7: 1; 13: 1; 29: 1 {{mpf le sqrt(n)|965536}} 2: 5, 11: 1, 13: 1, 211: 1 {{mpf le sqrt(n)|965537}} 67: 1 {{mpf le sqrt(n)|965539}} 83: 1 {{mpf le sqrt(n)|965541}} 3: 1 {{mpf le sqrt(n)|965543}} 383: 1 {{mpf le sqrt(n)|965547}} 3: 3, 11: 1 {{mpf le sqrt(n)|965549}} 13: 1, 17: 2, 257: 1 {{mpf le sqrt(n)|965551}} {{mpf le sqrt(n)|965553}} 3: 1 {{mpf le sqrt(n)|965557}} 31: 1 {{mpf le sqrt(n)|965559}} 3: 1, 7: 1 {{mpf le sqrt(n)|965561}} 19: 1, 89: 1, 571: 1 {{mpf le sqrt(n)|965563}} 23: 1 {{mpf le sqrt(n)|965567}} {{mpf le sqrt(n)|965569}} 11: 1, 61: 1 {{mpf le sqrt(n)|997 * 1019}} 997: 1 {{mpf le sqrt(n)|1015943}} 997: 1
Examples with invalid input
Code Result {{mpf le sqrt(n)|0}} Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer {{mpf le sqrt(n)|1031^2}} Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer with absolute value < 1031 2 = 1062961
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}}.[1]
code result {{prime factors (with multiplicity) up to sqrt(n)|1,000}} Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer {{prime factors (with multiplicity) up to sqrt(n)|{{formatnum:1,000|R}}}} 2: 3, 5: 3
Code
<noinclude>{{documentation}}<!--
A000040 The prime numbers.
The 172 primes less than 2^10 = 1024 are:
n a(n)
1 2
2 3
3 5
4 7
5 11
6 13
7 17
8 19
9 23
(...)
165 977
166 983
167 991
168 997
169 1009
170 1013
171 1019
172 1021
The next prime is:
173 1031
--></noinclude><includeonly>{{ifint| {{{1|NAN}}}
| {{#ifexpr: ( abs ({{{1}}}) ) < 1031^2
| {{#ifexpr: ( abs ({{{1}}}) ) > 1
| {{trim|<!--
A000040 The prime numbers.
The 172 primes less than 2^10 = 1024 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021
1 -->{{#ifexpr: ( ( 2 * 2) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 2 ) | 2{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 2}}{{{sep|{{{2|, }}}}}} }}<!--
2 -->{{#ifexpr: ( ( 3 * 3) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 3 ) | 3{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 3}}{{{sep|{{{2|, }}}}}} }}<!--
3 -->{{#ifexpr: ( ( 5 * 5) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 5 ) | 5{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 5}}{{{sep|{{{2|, }}}}}} }}<!--
4 -->{{#ifexpr: ( ( 7 * 7) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 7 ) | 7{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 7}}{{{sep|{{{2|, }}}}}} }}<!--
5 -->{{#ifexpr: ( ( 11 * 11) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 11 ) | 11{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 11}}{{{sep|{{{2|, }}}}}} }}<!--
6 -->{{#ifexpr: ( ( 13 * 13) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 13 ) | 13{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 13}}{{{sep|{{{2|, }}}}}} }}<!--
7 -->{{#ifexpr: ( ( 17 * 17) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 17 ) | 17{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 17}}{{{sep|{{{2|, }}}}}} }}<!--
8 -->{{#ifexpr: ( ( 19 * 19) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 19 ) | 19{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 19}}{{{sep|{{{2|, }}}}}} }}<!--
9 -->{{#ifexpr: ( ( 23 * 23) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 23 ) | 23{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 23}}{{{sep|{{{2|, }}}}}} }}<!--
10 -->{{#ifexpr: ( ( 29 * 29) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 29 ) | 29{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 29}}{{{sep|{{{2|, }}}}}} }}<!--
11 -->{{#ifexpr: ( ( 31 * 31) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 31 ) | 31{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 31}}{{{sep|{{{2|, }}}}}} }}<!--
12 -->{{#ifexpr: ( ( 37 * 37) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 37 ) | 37{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 37}}{{{sep|{{{2|, }}}}}} }}<!--
13 -->{{#ifexpr: ( ( 41 * 41) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 41 ) | 41{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 41}}{{{sep|{{{2|, }}}}}} }}<!--
14 -->{{#ifexpr: ( ( 43 * 43) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 43 ) | 43{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 43}}{{{sep|{{{2|, }}}}}} }}<!--
15 -->{{#ifexpr: ( ( 47 * 47) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 47 ) | 47{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 47}}{{{sep|{{{2|, }}}}}} }}<!--
16 -->{{#ifexpr: ( ( 53 * 53) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 53 ) | 53{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 53}}{{{sep|{{{2|, }}}}}} }}<!--
17 -->{{#ifexpr: ( ( 59 * 59) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 59 ) | 59{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 59}}{{{sep|{{{2|, }}}}}} }}<!--
18 -->{{#ifexpr: ( ( 61 * 61) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 61 ) | 61{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 61}}{{{sep|{{{2|, }}}}}} }}<!--
19 -->{{#ifexpr: ( ( 67 * 67) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 67 ) | 67{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 67}}{{{sep|{{{2|, }}}}}} }}<!--
20 -->{{#ifexpr: ( ( 71 * 71) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 71 ) | 71{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 71}}{{{sep|{{{2|, }}}}}} }}<!--
21 -->{{#ifexpr: ( ( 73 * 73) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 73 ) | 73{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 73}}{{{sep|{{{2|, }}}}}} }}<!--
22 -->{{#ifexpr: ( ( 79 * 79) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 79 ) | 79{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 79}}{{{sep|{{{2|, }}}}}} }}<!--
23 -->{{#ifexpr: ( ( 83 * 83) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 83 ) | 83{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 83}}{{{sep|{{{2|, }}}}}} }}<!--
24 -->{{#ifexpr: ( ( 89 * 89) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 89 ) | 89{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 89}}{{{sep|{{{2|, }}}}}} }}<!--
25 -->{{#ifexpr: ( ( 97 * 97) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 97 ) | 97{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}| 97}}{{{sep|{{{2|, }}}}}} }}<!--
26 -->{{#ifexpr: ( (101 * 101) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 101 ) | 101{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|101}}{{{sep|{{{2|, }}}}}} }}<!--
27 -->{{#ifexpr: ( (103 * 103) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 103 ) | 103{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|103}}{{{sep|{{{2|, }}}}}} }}<!--
28 -->{{#ifexpr: ( (107 * 107) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 107 ) | 107{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|107}}{{{sep|{{{2|, }}}}}} }}<!--
29 -->{{#ifexpr: ( (109 * 109) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 109 ) | 109{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|109}}{{{sep|{{{2|, }}}}}} }}<!--
30 -->{{#ifexpr: ( (113 * 113) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 113 ) | 113{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|113}}{{{sep|{{{2|, }}}}}} }}<!--
31 -->{{#ifexpr: ( (127 * 127) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 127 ) | 127{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|127}}{{{sep|{{{2|, }}}}}} }}<!--
-->{{#ifexpr: (131 * 131) <= abs ({{{1}}}) |<!--
32 -->{{#ifexpr: ( (131 * 131) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 131 ) | 131{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|131}}{{{sep|{{{2|, }}}}}} }}<!--
33 -->{{#ifexpr: ( (137 * 137) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 137 ) | 137{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|137}}{{{sep|{{{2|, }}}}}} }}<!--
34 -->{{#ifexpr: ( (139 * 139) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 139 ) | 139{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|139}}{{{sep|{{{2|, }}}}}} }}<!--
35 -->{{#ifexpr: ( (149 * 149) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 149 ) | 149{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|149}}{{{sep|{{{2|, }}}}}} }}<!--
36 -->{{#ifexpr: ( (151 * 151) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 151 ) | 151{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|151}}{{{sep|{{{2|, }}}}}} }}<!--
37 -->{{#ifexpr: ( (157 * 157) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 157 ) | 157{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|157}}{{{sep|{{{2|, }}}}}} }}<!--
38 -->{{#ifexpr: ( (163 * 163) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 163 ) | 163{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|163}}{{{sep|{{{2|, }}}}}} }}<!--
39 -->{{#ifexpr: ( (167 * 167) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 167 ) | 167{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|167}}{{{sep|{{{2|, }}}}}} }}<!--
40 -->{{#ifexpr: ( (173 * 173) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 173 ) | 173{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|173}}{{{sep|{{{2|, }}}}}} }}<!--
41 -->{{#ifexpr: ( (179 * 179) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 179 ) | 179{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|179}}{{{sep|{{{2|, }}}}}} }}<!--
42 -->{{#ifexpr: ( (181 * 181) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 181 ) | 181{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|181}}{{{sep|{{{2|, }}}}}} }}<!--
43 -->{{#ifexpr: ( (191 * 191) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 191 ) | 191{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|191}}{{{sep|{{{2|, }}}}}} }}<!--
44 -->{{#ifexpr: ( (193 * 193) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 193 ) | 193{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|193}}{{{sep|{{{2|, }}}}}} }}<!--
45 -->{{#ifexpr: ( (197 * 197) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 197 ) | 197{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|197}}{{{sep|{{{2|, }}}}}} }}<!--
46 -->{{#ifexpr: ( (199 * 199) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 199 ) | 199{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|199}}{{{sep|{{{2|, }}}}}} }}<!--
47 -->{{#ifexpr: ( (211 * 211) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 211 ) | 211{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|211}}{{{sep|{{{2|, }}}}}} }}<!--
48 -->{{#ifexpr: ( (223 * 223) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 223 ) | 223{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|223}}{{{sep|{{{2|, }}}}}} }}<!--
49 -->{{#ifexpr: ( (227 * 227) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 227 ) | 227{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|227}}{{{sep|{{{2|, }}}}}} }}<!--
50 -->{{#ifexpr: ( (229 * 229) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 229 ) | 229{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|229}}{{{sep|{{{2|, }}}}}} }}<!--
51 -->{{#ifexpr: ( (233 * 233) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 233 ) | 233{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|233}}{{{sep|{{{2|, }}}}}} }}<!--
52 -->{{#ifexpr: ( (239 * 239) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 239 ) | 239{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|239}}{{{sep|{{{2|, }}}}}} }}<!--
53 -->{{#ifexpr: ( (241 * 241) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 241 ) | 241{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|241}}{{{sep|{{{2|, }}}}}} }}<!--
54 -->{{#ifexpr: ( (251 * 251) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 251 ) | 251{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|251}}{{{sep|{{{2|, }}}}}} }}<!--
A000040 The prime numbers.
The 172 primes less than 2^10 = 1024 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021
-->}}{{#ifexpr: (257 * 257) <= abs ({{{1}}}) |<!--
55 -->{{#ifexpr: ( (257 * 257) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 257 ) | 257{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|257}}{{{sep|{{{2|, }}}}}} }}<!--
56 -->{{#ifexpr: ( (263 * 263) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 263 ) | 263{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|263}}{{{sep|{{{2|, }}}}}} }}<!--
57 -->{{#ifexpr: ( (269 * 269) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 269 ) | 269{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|269}}{{{sep|{{{2|, }}}}}} }}<!--
58 -->{{#ifexpr: ( (271 * 271) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 271 ) | 271{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|271}}{{{sep|{{{2|, }}}}}} }}<!--
59 -->{{#ifexpr: ( (277 * 277) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 277 ) | 277{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|277}}{{{sep|{{{2|, }}}}}} }}<!--
60 -->{{#ifexpr: ( (281 * 281) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 281 ) | 281{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|281}}{{{sep|{{{2|, }}}}}} }}<!--
61 -->{{#ifexpr: ( (283 * 283) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 283 ) | 283{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|283}}{{{sep|{{{2|, }}}}}} }}<!--
62 -->{{#ifexpr: ( (293 * 293) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 293 ) | 293{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|293}}{{{sep|{{{2|, }}}}}} }}<!--
63 -->{{#ifexpr: ( (307 * 307) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 307 ) | 307{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|307}}{{{sep|{{{2|, }}}}}} }}<!--
64 -->{{#ifexpr: ( (311 * 311) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 311 ) | 311{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|311}}{{{sep|{{{2|, }}}}}} }}<!--
65 -->{{#ifexpr: ( (313 * 313) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 313 ) | 313{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|313}}{{{sep|{{{2|, }}}}}} }}<!--
66 -->{{#ifexpr: ( (317 * 317) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 317 ) | 317{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|317}}{{{sep|{{{2|, }}}}}} }}<!--
67 -->{{#ifexpr: ( (331 * 331) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 331 ) | 331{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|331}}{{{sep|{{{2|, }}}}}} }}<!--
68 -->{{#ifexpr: ( (337 * 337) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 337 ) | 337{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|337}}{{{sep|{{{2|, }}}}}} }}<!--
69 -->{{#ifexpr: ( (347 * 347) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 347 ) | 347{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|347}}{{{sep|{{{2|, }}}}}} }}<!--
70 -->{{#ifexpr: ( (349 * 349) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 349 ) | 349{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|349}}{{{sep|{{{2|, }}}}}} }}<!--
71 -->{{#ifexpr: ( (353 * 353) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 353 ) | 353{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|353}}{{{sep|{{{2|, }}}}}} }}<!--
72 -->{{#ifexpr: ( (359 * 359) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 359 ) | 359{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|359}}{{{sep|{{{2|, }}}}}} }}<!--
73 -->{{#ifexpr: ( (367 * 367) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 367 ) | 367{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|367}}{{{sep|{{{2|, }}}}}} }}<!--
74 -->{{#ifexpr: ( (373 * 373) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 373 ) | 373{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|373}}{{{sep|{{{2|, }}}}}} }}<!--
75 -->{{#ifexpr: ( (379 * 379) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 379 ) | 379{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|379}}{{{sep|{{{2|, }}}}}} }}<!--
76 -->{{#ifexpr: ( (383 * 383) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 383 ) | 383{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|383}}{{{sep|{{{2|, }}}}}} }}<!--
77 -->{{#ifexpr: ( (389 * 389) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 389 ) | 389{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|389}}{{{sep|{{{2|, }}}}}} }}<!--
78 -->{{#ifexpr: ( (397 * 397) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 397 ) | 397{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|397}}{{{sep|{{{2|, }}}}}} }}<!--
79 -->{{#ifexpr: ( (401 * 401) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 401 ) | 401{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|401}}{{{sep|{{{2|, }}}}}} }}<!--
80 -->{{#ifexpr: ( (409 * 409) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 409 ) | 409{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|409}}{{{sep|{{{2|, }}}}}} }}<!--
81 -->{{#ifexpr: ( (419 * 419) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 419 ) | 419{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|419}}{{{sep|{{{2|, }}}}}} }}<!--
82 -->{{#ifexpr: ( (421 * 421) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 421 ) | 421{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|421}}{{{sep|{{{2|, }}}}}} }}<!--
83 -->{{#ifexpr: ( (431 * 431) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 431 ) | 431{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|431}}{{{sep|{{{2|, }}}}}} }}<!--
84 -->{{#ifexpr: ( (433 * 433) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 433 ) | 433{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|433}}{{{sep|{{{2|, }}}}}} }}<!--
85 -->{{#ifexpr: ( (439 * 439) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 439 ) | 439{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|439}}{{{sep|{{{2|, }}}}}} }}<!--
86 -->{{#ifexpr: ( (443 * 443) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 443 ) | 443{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|443}}{{{sep|{{{2|, }}}}}} }}<!--
87 -->{{#ifexpr: ( (449 * 449) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 449 ) | 449{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|449}}{{{sep|{{{2|, }}}}}} }}<!--
88 -->{{#ifexpr: ( (457 * 457) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 457 ) | 457{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|457}}{{{sep|{{{2|, }}}}}} }}<!--
89 -->{{#ifexpr: ( (461 * 461) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 461 ) | 461{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|461}}{{{sep|{{{2|, }}}}}} }}<!--
90 -->{{#ifexpr: ( (463 * 463) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 463 ) | 463{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|463}}{{{sep|{{{2|, }}}}}} }}<!--
91 -->{{#ifexpr: ( (467 * 467) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 467 ) | 467{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|467}}{{{sep|{{{2|, }}}}}} }}<!--
92 -->{{#ifexpr: ( (479 * 479) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 479 ) | 479{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|479}}{{{sep|{{{2|, }}}}}} }}<!--
93 -->{{#ifexpr: ( (487 * 487) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 487 ) | 487{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|487}}{{{sep|{{{2|, }}}}}} }}<!--
94 -->{{#ifexpr: ( (491 * 491) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 491 ) | 491{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|491}}{{{sep|{{{2|, }}}}}} }}<!--
95 -->{{#ifexpr: ( (499 * 499) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 499 ) | 499{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|499}}{{{sep|{{{2|, }}}}}} }}<!--
96 -->{{#ifexpr: ( (503 * 503) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 503 ) | 503{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|503}}{{{sep|{{{2|, }}}}}} }}<!--
97 -->{{#ifexpr: ( (509 * 509) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 509 ) | 509{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|509}}{{{sep|{{{2|, }}}}}} }}<!--
A000040 The prime numbers.
The 172 primes less than 2^10 = 1024 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021
-->}}{{#ifexpr: (521 * 521) <= abs ({{{1}}}) |<!--
98 -->{{#ifexpr: ( (521 * 521) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 521 ) | 521{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|521}}{{{sep|{{{2|, }}}}}} }}<!--
99 -->{{#ifexpr: ( (523 * 523) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 523 ) | 523{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|523}}{{{sep|{{{2|, }}}}}} }}<!--
100 -->{{#ifexpr: ( (541 * 541) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 541 ) | 541{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|541}}{{{sep|{{{2|, }}}}}} }}<!--
101 -->{{#ifexpr: ( (547 * 547) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 547 ) | 547{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|547}}{{{sep|{{{2|, }}}}}} }}<!--
102 -->{{#ifexpr: ( (557 * 557) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 557 ) | 557{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|557}}{{{sep|{{{2|, }}}}}} }}<!--
103 -->{{#ifexpr: ( (563 * 563) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 563 ) | 563{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|563}}{{{sep|{{{2|, }}}}}} }}<!--
104 -->{{#ifexpr: ( (569 * 569) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 569 ) | 569{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|569}}{{{sep|{{{2|, }}}}}} }}<!--
105 -->{{#ifexpr: ( (571 * 571) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 571 ) | 571{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|571}}{{{sep|{{{2|, }}}}}} }}<!--
106 -->{{#ifexpr: ( (577 * 577) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 577 ) | 577{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|577}}{{{sep|{{{2|, }}}}}} }}<!--
107 -->{{#ifexpr: ( (587 * 587) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 587 ) | 587{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|587}}{{{sep|{{{2|, }}}}}} }}<!--
108 -->{{#ifexpr: ( (593 * 593) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 593 ) | 593{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|593}}{{{sep|{{{2|, }}}}}} }}<!--
109 -->{{#ifexpr: ( (599 * 599) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 599 ) | 599{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|599}}{{{sep|{{{2|, }}}}}} }}<!--
110 -->{{#ifexpr: ( (601 * 601) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 601 ) | 601{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|601}}{{{sep|{{{2|, }}}}}} }}<!--
111 -->{{#ifexpr: ( (607 * 607) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 607 ) | 607{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|607}}{{{sep|{{{2|, }}}}}} }}<!--
112 -->{{#ifexpr: ( (613 * 613) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 613 ) | 613{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|613}}{{{sep|{{{2|, }}}}}} }}<!--
113 -->{{#ifexpr: ( (617 * 617) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 617 ) | 617{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|617}}{{{sep|{{{2|, }}}}}} }}<!--
114 -->{{#ifexpr: ( (619 * 619) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 619 ) | 619{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|619}}{{{sep|{{{2|, }}}}}} }}<!--
115 -->{{#ifexpr: ( (631 * 631) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 631 ) | 631{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|631}}{{{sep|{{{2|, }}}}}} }}<!--
116 -->{{#ifexpr: ( (641 * 641) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 641 ) | 641{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|641}}{{{sep|{{{2|, }}}}}} }}<!--
117 -->{{#ifexpr: ( (643 * 643) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 643 ) | 643{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|643}}{{{sep|{{{2|, }}}}}} }}<!--
118 -->{{#ifexpr: ( (647 * 647) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 647 ) | 647{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|647}}{{{sep|{{{2|, }}}}}} }}<!--
119 -->{{#ifexpr: ( (653 * 653) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 653 ) | 653{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|653}}{{{sep|{{{2|, }}}}}} }}<!--
120 -->{{#ifexpr: ( (659 * 659) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 659 ) | 659{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|659}}{{{sep|{{{2|, }}}}}} }}<!--
121 -->{{#ifexpr: ( (661 * 661) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 661 ) | 661{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|661}}{{{sep|{{{2|, }}}}}} }}<!--
122 -->{{#ifexpr: ( (673 * 673) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 673 ) | 673{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|673}}{{{sep|{{{2|, }}}}}} }}<!--
123 -->{{#ifexpr: ( (677 * 677) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 677 ) | 677{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|677}}{{{sep|{{{2|, }}}}}} }}<!--
124 -->{{#ifexpr: ( (683 * 683) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 683 ) | 683{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|683}}{{{sep|{{{2|, }}}}}} }}<!--
125 -->{{#ifexpr: ( (691 * 691) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 691 ) | 691{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|691}}{{{sep|{{{2|, }}}}}} }}<!--
126 -->{{#ifexpr: ( (701 * 701) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 701 ) | 701{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|701}}{{{sep|{{{2|, }}}}}} }}<!--
127 -->{{#ifexpr: ( (709 * 709) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 709 ) | 709{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|709}}{{{sep|{{{2|, }}}}}} }}<!--
128 -->{{#ifexpr: ( (719 * 719) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 719 ) | 719{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|719}}{{{sep|{{{2|, }}}}}} }}<!--
129 -->{{#ifexpr: ( (727 * 727) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 727 ) | 727{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|727}}{{{sep|{{{2|, }}}}}} }}<!--
130 -->{{#ifexpr: ( (733 * 733) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 733 ) | 733{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|733}}{{{sep|{{{2|, }}}}}} }}<!--
131 -->{{#ifexpr: ( (739 * 739) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 739 ) | 739{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|739}}{{{sep|{{{2|, }}}}}} }}<!--
132 -->{{#ifexpr: ( (743 * 743) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 743 ) | 743{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|743}}{{{sep|{{{2|, }}}}}} }}<!--
133 -->{{#ifexpr: ( (751 * 751) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 751 ) | 751{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|751}}{{{sep|{{{2|, }}}}}} }}<!--
134 -->{{#ifexpr: ( (757 * 757) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 757 ) | 757{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|757}}{{{sep|{{{2|, }}}}}} }}<!--
135 -->{{#ifexpr: ( (761 * 761) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 761 ) | 761{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|761}}{{{sep|{{{2|, }}}}}} }}<!--
A000040 The prime numbers.
The 172 primes less than 2^10 = 1024 are:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283,293,307,311,313,317,331,337,347,349,353,359,367,373,379,383,389,397,401,409,419,421,431,433,439,443,449,457,461,463,467,479,487,491,499,503,509,521,523,541,547,557,563,569,571,577,587,593,599,601,607,613,617,619,631,641,643,647,653,659,661,673,677,683,691,701,709,719,727,733,739,743,751,757,761,769,773,787,797,809,811,821,823,827,829,839,853,857,859,863,877,881,883,887,907,911,919,929,937,941,947,953,967,971,977,983,991,997,1009,1013,1019,1021
-->}}{{#ifexpr: (769 * 769) <= abs ({{{1}}}) |<!--
136 -->{{#ifexpr: ( (769 * 769) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 769 ) | 769{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|769}}{{{sep|{{{2|, }}}}}} }}<!--
137 -->{{#ifexpr: ( (773 * 773) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 773 ) | 773{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|773}}{{{sep|{{{2|, }}}}}} }}<!--
138 -->{{#ifexpr: ( (787 * 787) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 787 ) | 787{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|787}}{{{sep|{{{2|, }}}}}} }}<!--
139 -->{{#ifexpr: ( (797 * 797) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 797 ) | 797{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|797}}{{{sep|{{{2|, }}}}}} }}<!--
140 -->{{#ifexpr: ( (809 * 809) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 809 ) | 809{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|809}}{{{sep|{{{2|, }}}}}} }}<!--
141 -->{{#ifexpr: ( (811 * 811) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 811 ) | 811{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|811}}{{{sep|{{{2|, }}}}}} }}<!--
142 -->{{#ifexpr: ( (821 * 821) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 821 ) | 821{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|821}}{{{sep|{{{2|, }}}}}} }}<!--
143 -->{{#ifexpr: ( (823 * 823) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 823 ) | 823{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|823}}{{{sep|{{{2|, }}}}}} }}<!--
144 -->{{#ifexpr: ( (827 * 827) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 827 ) | 827{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|827}}{{{sep|{{{2|, }}}}}} }}<!--
145 -->{{#ifexpr: ( (829 * 829) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 829 ) | 829{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|829}}{{{sep|{{{2|, }}}}}} }}<!--
146 -->{{#ifexpr: ( (839 * 839) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 839 ) | 839{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|839}}{{{sep|{{{2|, }}}}}} }}<!--
147 -->{{#ifexpr: ( (853 * 853) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 853 ) | 853{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|853}}{{{sep|{{{2|, }}}}}} }}<!--
148 -->{{#ifexpr: ( (857 * 857) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 857 ) | 857{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|857}}{{{sep|{{{2|, }}}}}} }}<!--
149 -->{{#ifexpr: ( (859 * 859) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 859 ) | 859{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|859}}{{{sep|{{{2|, }}}}}} }}<!--
150 -->{{#ifexpr: ( (863 * 863) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 863 ) | 863{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|863}}{{{sep|{{{2|, }}}}}} }}<!--
151 -->{{#ifexpr: ( (877 * 877) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 877 ) | 877{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|877}}{{{sep|{{{2|, }}}}}} }}<!--
152 -->{{#ifexpr: ( (881 * 881) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 881 ) | 881{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|881}}{{{sep|{{{2|, }}}}}} }}<!--
153 -->{{#ifexpr: ( (883 * 883) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 883 ) | 883{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|883}}{{{sep|{{{2|, }}}}}} }}<!--
154 -->{{#ifexpr: ( (887 * 887) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 887 ) | 887{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|887}}{{{sep|{{{2|, }}}}}} }}<!--
155 -->{{#ifexpr: ( (907 * 907) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 907 ) | 907{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|907}}{{{sep|{{{2|, }}}}}} }}<!--
156 -->{{#ifexpr: ( (911 * 911) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 911 ) | 911{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|911}}{{{sep|{{{2|, }}}}}} }}<!--
157 -->{{#ifexpr: ( (919 * 919) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 919 ) | 919{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|919}}{{{sep|{{{2|, }}}}}} }}<!--
158 -->{{#ifexpr: ( (929 * 929) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 929 ) | 929{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|929}}{{{sep|{{{2|, }}}}}} }}<!--
159 -->{{#ifexpr: ( (937 * 937) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 937 ) | 937{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|937}}{{{sep|{{{2|, }}}}}} }}<!--
160 -->{{#ifexpr: ( (941 * 941) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 941 ) | 941{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|941}}{{{sep|{{{2|, }}}}}} }}<!--
161 -->{{#ifexpr: ( (947 * 947) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 947 ) | 947{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|947}}{{{sep|{{{2|, }}}}}} }}<!--
162 -->{{#ifexpr: ( (953 * 953) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 953 ) | 953{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|953}}{{{sep|{{{2|, }}}}}} }}<!--
163 -->{{#ifexpr: ( (967 * 967) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 967 ) | 967{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|967}}{{{sep|{{{2|, }}}}}} }}<!--
164 -->{{#ifexpr: ( (971 * 971) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 971 ) | 971{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|971}}{{{sep|{{{2|, }}}}}} }}<!--
165 -->{{#ifexpr: ( (977 * 977) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 977 ) | 977{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|977}}{{{sep|{{{2|, }}}}}} }}<!--
166 -->{{#ifexpr: ( (983 * 983) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 983 ) | 983{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|983}}{{{sep|{{{2|, }}}}}} }}<!--
167 -->{{#ifexpr: ( (991 * 991) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 991 ) | 991{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|991}}{{{sep|{{{2|, }}}}}} }}<!--
168 -->{{#ifexpr: ( (997 * 997) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 997 ) | 997{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|997}}{{{sep|{{{2|, }}}}}} }}<!--
169 -->{{#ifexpr: ( (1009 * 1009) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 1009 ) | 1009{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|1009}}{{{sep|{{{2|, }}}}}} }}<!--
170 -->{{#ifexpr: ( (1013 * 1013) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 1013 ) | 1013{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|1013}}{{{sep|{{{2|, }}}}}} }}<!--
171 -->{{#ifexpr: ( (1019 * 1019) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 1019 ) | 1019{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|1019}}{{{sep|{{{2|, }}}}}} }}<!--
172 -->{{#ifexpr: ( (1021 * 1021) <= abs ({{{1}}}) ) and not ( abs ({{{1}}}) mod 1021 ) | 1021{{{key/val_sep|{{{3|: }}}}}}{{multiplicity|{{{1}}}|1021}}{{{sep|{{{2|, }}}}}} }}<!--
-->}}<!--
-->|{{{sep|{{{2|, }}}}}}<!--
-->}}
| {{#ifeq: {{{1}}} | 0
| {{error| Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer }}
| <!-- Empty list for -1 or 1 (no prime factors) -->
}}
}}
| {{error| Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer with absolute value < 1031{{^|2}} {{=}} {{#expr: 1031^2}} }}
}}
| {{error| Prime factors (with multiplicity) up to sqrt(n) error: Argument must be a nonzero integer }}
}}</includeonly>
See also
- {{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
(default value))k = 1 - {{divisor function}} or {{sigma k}} (for
)k ≠ 0
External links
- Andrew Hodges, Java Applet for Factorization
- http://factordb.com/
