Table Of Prime Factors

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* Properties
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* 1 to 100
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* 101 to 200
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* 201 to 300
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* 301 to 400
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* 401 to 500
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* 501 to 600
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* 601 to 700
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* 701 to 800
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* 801 to 900
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* 901 to 1000
*
* See also
*
Table of prime factors
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From Wikipedia, the free encyclopedia
The tables contain the prime factorization of the natural numbers from 1 to 1000.
When n is a prime number, the prime factorization is just n itself, written in bold below.
The number 1 is called a unit. It has no prime factors and is neither prime nor composite.
Properties
Many properties of a natural number n can be seen or directly computed from the prime factorization of n.
* The multiplicity of a prime factor p of n is the largest exponent m for which pm divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p1). The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
* Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities).
* A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers.
* A composite number has Ω(n) > 1. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 (sequence A002808 in the OEIS). All numbers above 1 are either prime or composite. 1 is neither.
* A semiprime has Ω(n) = 2 (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 (sequence A001358 in the OEIS).
* A k-almost prime (for a natural number k) has Ω(n) = k (so it is composite if k > 1).
* An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (sequence A005843 in the OEIS).
* An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd.
* A square has even multiplicity for all prime factors (it is of the form a2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS).
* A cube has all multiplicities divisible by 3 (it is of the form a3 for some a). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS).
* A perfect power has a common divisor m > 1 for all multiplicities (it is of the form am for some a > 1 and m > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included.
* A powerful number (also called squareful) has multiplicity above 1 for all prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 (sequence A001694 in the OEIS).
* A prime power has only one prime factor. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 (sequence A000961 in the OEIS). 1 is sometimes included.
* An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968 (sequence A052486 in the OEIS).
* A square-free integer has no prime factor with multiplicity above 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 (sequence A005117 in the OEIS). A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful.
* The Liouville function λ(n) is 1 if Ω(n) is even, and is -1 if Ω(n) is odd.
* The Möbius function μ(n) is 0 if n is not square-free. Otherwise μ(n) is 1 if Ω(n) is even, and is −1 if Ω(n) is odd.
* A sphenic number has Ω(n) = 3 and is square-free (so it is the product of 3 distinct primes). The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 (sequence A007304 in the OEIS).
* a0(n) is the sum of primes dividing n, counted with multiplicity. It is an additive function.
* A Ruth-Aaron pair is two consecutive numbers (x, x+1) with a0(x) = a0(x+1). The first (by x value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 (sequence A039752 in the OEIS). Another definition is the same prime only counted once; if so, the first (by x value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 (sequence A006145 in the OEIS).
* A primorial x# is the product of all primes from 2 to x. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 (sequence A002110 in the OEIS). 1# = 1 is sometimes included.
* A factorial x! is the product of all numbers from 1 to x. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included.
* A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k).
* m is smoother than n if the largest prime factor of m is below the largest of n.
* A regular number has no prime factor above 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 (sequence A051037 in the OEIS).
* A k-powersmooth number has all pm ≤ k where p is a prime factor with multiplicity m.
* A frugal number has more digits than the number of digits in its prime factorization (when written like the tables below with multiplicities above 1 as exponents). The first in decimal: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 (sequence A046759 in the OEIS).
* An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17 (sequence A046758 in the OEIS).
* An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS).
* An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
* gcd(m, n) (greatest common divisor of m and n) is the product of all prime factors which are both in m and n (with the smallest multiplicity for m and n).
* m and n are coprime (also called relatively prime) if gcd(m, n) = 1 (meaning they have no common prime factor).
* lcm(m, n) (least common multiple of m and n) is the product of all prime factors of m or n (with the largest multiplicity for m or n).
* gcd(m, n) × lcm(m, n) = m × n. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization.
* m is a divisor of n (also called m divides n, or n is divisible by m) if all prime factors of m have at least the same multiplicity in n.
The divisors of n are all products of some or all prime factors of n (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.
1 to 100
1 − 20
{| class="fandom-table"	1 	
2 	2 	
3 	3 	
4 	22 	
5 	5 	
6 	2·3 	
7 	7 	
8 	23 	
9 	32 	
10	2·5 	
11	11 	
12	22·3	
13	13 	
14	2·7 	
15	3·5 	
16	24 	
17	17 	
18	2·32	
19	19 	
20	22·5	
101 to 200
101 − 120
{| class="fandom-table"	101 	101
102	2·3·17	
103	103 	
104	23·13 	
105	3·5·7 	
106	2·53 	
107	107 	
108	22·33 	
109	109 	
110	2·5·11	
111	3·37 	
112	24·7 	
113	113 	
114	2·3·19	
115	5·23 	
116	22·29 	
117	32·13 	
118	2·59 	
119	7·17 	
120	23·3·5	
201 to 300
201 − 220
{| class="fandom-table"	201 	3·67
202	2·101 	
203	7·29 	
204	22·3·17	
205	5·41 	
206	2·103 	
207	32·23 	
208	24·13 	
209	11·19 	
210	2·3·5·7	
211	211 	
212	22·53 	
213	3·71 	
214	2·107 	
215	5·43 	
216	23·33 	
217	7·31 	
218	2·109 	
219	3·73 	
220	22·5·11	
301 to 400
301 − 320
{| class="fandom-table"	301 	7·43
302	2·151 	
303	3·101 	
304	24·19 	
305	5·61 	
306	2·32·17	
307	307 	
308	22·7·11	
309	3·103 	
310	2·5·31 	
311	311 	
312	23·3·13	
313	313 	
314	2·157 	
315	32·5·7 	
316	22·79 	
317	317 	
318	2·3·53 	
319	11·29 	
320	26·5 	
401 to 500
401 − 420
{| class="fandom-table"	401 	401
402	2·3·67 	
403	13·31 	
404	22·101 	
405	34·5 	
406	2·7·29 	
407	11·37 	
408	23·3·17 	
409	409 	
410	2·5·41 	
411	3·137 	
412	22·103 	
413	7·59 	
414	2·32·23 	
415	5·83 	
416	25·13 	
417	3·139 	
418	2·11·19 	
419	419 	
420	22·3·5·7	
501 to 600
501 − 520
{| class="fandom-table"	501 	3·167
502	2·251 	
503	503 	
504	23·32·7 	
505	5·101 	
506	2·11·23 	
507	3·132 	
508	22·127 	
509	509 	
510	2·3·5·17	
511	7·73 	
512	29 	
513	33·19 	
514	2·257 	
515	5·103 	
516	22·3·43 	
517	11·47 	
518	2·7·37 	
519	3·173 	
520	23·5·13 	
601 to 700
601 − 620
{| class="fandom-table"	601 	601
602	2·7·43 	
603	32·67 	
604	22·151 	
605	5·112 	
606	2·3·101 	
607	607 	
608	25·19 	
609	3·7·29 	
610	2·5·61 	
611	13·47 	
612	22·32·17	
613	613 	
614	2·307 	
615	3·5·41 	
616	23·7·11 	
617	617 	
618	2·3·103 	
619	619 	
620	22·5·31 	
701 to 
800
701 − 720
{| class="fandom-table"	701 	701
702	2·33·13 	
703	19·37 	
704	26·11 	
705	3·5·47 	
706	2·353 	
707	7·101 	
708	22·3·59 	
709	709 	
710	2·5·71 	
711	32·79 	
712	23·89 	
713	23·31 	
714	2·3·7·17	
715	5·11·13 	
716	22·179 	
717	3·239 	
718	2·359 	
719	719 	
720	24·32·5 	
801 to 900
801 - 820
{| class="fandom-table"	801 	32·89
802	2·401 	
803	11·73 	
804	22·3·67	
805	5·7·23 	
806	2·13·31	
807	3·269 	
808	23·101 	
809	809 	
810	2·34·5 	
811	811 	
812	22·7·29	
813	3·271 	
814	2·11·37	
815	5·163 	
816	24·3·17	
817	19·43 	
818	2·409 	
819	32·7·13	
820	22·5·41	
901 to 1000
901 - 920
{| class="fandom-table"	901 	17·53
902	2·11·41 	
903	3·7·43 	
904	23·113 	
905	5·181 	
906	2·3·151 	
907	907 	
908	22·227 	
909	32·101 	
910	2·5·7·13	
911	911 	
912	24·3·19 	
913	11·83 	
914	2·457 	
915	3·5·61 	
916	22·229 	
917	7·131 	
918	2·33·17 	
919	919 	
920	23·5·23 	
See also
* Fundamental theorem of arithmetic – Integers have unique prime factorizations
* List of prime numbers – List of prime numbers and notable types of prime numbers
* Table of divisors
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* Categories: Prime numbersElementary number theoryMathematics-related listsMathematical tablesNumber-related lists
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