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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.
Usage
[edit]
{{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
[edit]A nonzero integer less than 1031 2 = 1062961.
Examples
[edit]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
[edit]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
[edit]
<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
[edit]
- {{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
[edit]- Andrew Hodges, Java Applet for Factorization
- http://factordb.com/
