Table of prime factors

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.

See also: Table of divisors (prime and non-prime divisors for 1 to 1000)

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 big 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 count 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 largest prime factor ≤ 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 pmk 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 below tables 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
1
22
33
422
55
62·3
77
823
932
102·5
1111
1222·3
1313
142·7
153·5
1624
1717
182·32
1919
2022·5
21 − 40
213·7
222·11
2323
2423·3
2552
262·13
2733
2822·7
2929
302·3·5
3131
3225
333·11
342·17
355·7
3622·32
3737
382·19
393·13
4023·5
41 − 60
4141
422·3·7
4343
4422·11
4532·5
462·23
4747
4824·3
4972
502·52
513·17
5222·13
5353
542·33
555·11
5623·7
573·19
582·29
5959
6022·3·5
61 − 80
6161
622·31
6332·7
6426
655·13
662·3·11
6767
6822·17
693·23
702·5·7
7171
7223·32
7373
742·37
753·52
7622·19
777·11
782·3·13
7979
8024·5
81 − 100
8134
822·41
8383
8422·3·7
855·17
862·43
873·29
8823·11
8989
902·32·5
917·13
9222·23
933·31
942·47
955·19
9625·3
9797
982·72
9932·11
10022·52

101 to 200

101 − 120
101101
1022·3·17
103103
10423·13
1053·5·7
1062·53
107107
10822·33
109109
1102·5·11
1113·37
11224·7
113113
1142·3·19
1155·23
11622·29
11732·13
1182·59
1197·17
12023·3·5
121 − 140
121112
1222·61
1233·41
12422·31
12553
1262·32·7
127127
12827
1293·43
1302·5·13
131131
13222·3·11
1337·19
1342·67
13533·5
13623·17
137137
1382·3·23
139139
14022·5·7
141 − 160
1413·47
1422·71
14311·13
14424·32
1455·29
1462·73
1473·72
14822·37
149149
1502·3·52
151151
15223·19
15332·17
1542·7·11
1555·31
15622·3·13
157157
1582·79
1593·53
16025·5
161 − 180
1617·23
1622·34
163163
16422·41
1653·5·11
1662·83
167167
16823·3·7
169132
1702·5·17
17132·19
17222·43
173173
1742·3·29
17552·7
17624·11
1773·59
1782·89
179179
18022·32·5
181 − 200
181181
1822·7·13
1833·61
18423·23
1855·37
1862·3·31
18711·17
18822·47
18933·7
1902·5·19
191191
19226·3
193193
1942·97
1953·5·13
19622·72
197197
1982·32·11
199199
20023·52

201 to 300

201 − 220
2013·67
2022·101
2037·29
20422·3·17
2055·41
2062·103
20732·23
20824·13
20911·19
2102·3·5·7
211211
21222·53
2133·71
2142·107
2155·43
21623·33
2177·31
2182·109
2193·73
22022·5·11
221 − 240
22113·17
2222·3·37
223223
22425·7
22532·52
2262·113
227227
22822·3·19
229229
2302·5·23
2313·7·11
23223·29
233233
2342·32·13
2355·47
23622·59
2373·79
2382·7·17
239239
24024·3·5
241 − 260
241241
2422·112
24335
24422·61
2455·72
2462·3·41
24713·19
24823·31
2493·83
2502·53
251251
25222·32·7
25311·23
2542·127
2553·5·17
25628
257257
2582·3·43
2597·37
26022·5·13
261 − 280
26132·29
2622·131
263263
26423·3·11
2655·53
2662·7·19
2673·89
26822·67
269269
2702·33·5
271271
27224·17
2733·7·13
2742·137
27552·11
27622·3·23
277277
2782·139
27932·31
28023·5·7
281 − 300
281281
2822·3·47
283283
28422·71
2853·5·19
2862·11·13
2877·41
28825·32
289172
2902·5·29
2913·97
29222·73
293293
2942·3·72
2955·59
29623·37
29733·11
2982·149
29913·23
30022·3·52

301 to 400

301 − 320
3017·43
3022·151
3033·101
30424·19
3055·61
3062·32·17
307307
30822·7·11
3093·103
3102·5·31
311311
31223·3·13
313313
3142·157
31532·5·7
31622·79
317317
3182·3·53
31911·29
32026·5
321 − 340
3213·107
3222·7·23
32317·19
32422·34
32552·13
3262·163
3273·109
32823·41
3297·47
3302·3·5·11
331331
33222·83
33332·37
3342·167
3355·67
33624·3·7
337337
3382·132
3393·113
34022·5·17
341 − 360
34111·31
3422·32·19
34373
34423·43
3453·5·23
3462·173
347347
34822·3·29
349349
3502·52·7
35133·13
35225·11
353353
3542·3·59
3555·71
35622·89
3573·7·17
3582·179
359359
36023·32·5
361 − 380
361192
3622·181
3633·112
36422·7·13
3655·73
3662·3·61
367367
36824·23
36932·41
3702·5·37
3717·53
37222·3·31
373373
3742·11·17
3753·53
37623·47
37713·29
3782·33·7
379379
38022·5·19
381 − 400
3813·127
3822·191
383383
38427·3
3855·7·11
3862·193
38732·43
38822·97
389389
3902·3·5·13
39117·23
39223·72
3933·131
3942·197
3955·79
39622·32·11
397397
3982·199
3993·7·19
40024·52

401 to 500

401 − 420
401401
4022·3·67
40313·31
40422·101
40534·5
4062·7·29
40711·37
40823·3·17
409409
4102·5·41
4113·137
41222·103
4137·59
4142·32·23
4155·83
41625·13
4173·139
4182·11·19
419419
42022·3·5·7
421 − 440
421421
4222·211
42332·47
42423·53
42552·17
4262·3·71
4277·61
42822·107
4293·11·13
4302·5·43
431431
43224·33
433433
4342·7·31
4353·5·29
43622·109
43719·23
4382·3·73
439439
44023·5·11
441 − 460
44132·72
4422·13·17
443443
44422·3·37
4455·89
4462·223
4473·149
44826·7
449449
4502·32·52
45111·41
45222·113
4533·151
4542·227
4555·7·13
45623·3·19
457457
4582·229
45933·17
46022·5·23
461 − 480
461461
4622·3·7·11
463463
46424·29
4653·5·31
4662·233
467467
46822·32·13
4697·67
4702·5·47
4713·157
47223·59
47311·43
4742·3·79
47552·19
47622·7·17
47732·53
4782·239
479479
48025·3·5
481 − 500
48113·37
4822·241
4833·7·23
48422·112
4855·97
4862·35
487487
48823·61
4893·163
4902·5·72
491491
49222·3·41
49317·29
4942·13·19
49532·5·11
49624·31
4977·71
4982·3·83
499499
50022·53

501 to 600

501 − 520
5013·167
5022·251
503503
50423·32·7
5055·101
5062·11·23
5073·132
50822·127
509509
5102·3·5·17
5117·73
51229
51333·19
5142·257
5155·103
51622·3·43
51711·47
5182·7·37
5193·173
52023·5·13
521 − 540
521521
5222·32·29
523523
52422·131
5253·52·7
5262·263
52717·31
52824·3·11
529232
5302·5·53
53132·59
53222·7·19
53313·41
5342·3·89
5355·107
53623·67
5373·179
5382·269
53972·11
54022·33·5
541 − 560
541541
5422·271
5433·181
54425·17
5455·109
5462·3·7·13
547547
54822·137
54932·61
5502·52·11
55119·29
55223·3·23
5537·79
5542·277
5553·5·37
55622·139
557557
5582·32·31
55913·43
56024·5·7
561 − 580
5613·11·17
5622·281
563563
56422·3·47
5655·113
5662·283
56734·7
56823·71
569569
5702·3·5·19
571571
57222·11·13
5733·191
5742·7·41
57552·23
57626·32
577577
5782·172
5793·193
58022·5·29
581 − 600
5817·83
5822·3·97
58311·53
58423·73
58532·5·13
5862·293
587587
58822·3·72
58919·31
5902·5·59
5913·197
59224·37
593593
5942·33·11
5955·7·17
59622·149
5973·199
5982·13·23
599599
60023·3·52

601 to 700

601 − 620
601601
6022·7·43
60332·67
60422·151
6055·112
6062·3·101
607607
60825·19
6093·7·29
6102·5·61
61113·47
61222·32·17
613613
6142·307
6153·5·41
61623·7·11
617617
6182·3·103
619619
62022·5·31
621 − 640
62133·23
6222·311
6237·89
62424·3·13
62554
6262·313
6273·11·19
62822·157
62917·37
6302·32·5·7
631631
63223·79
6333·211
6342·317
6355·127
63622·3·53
63772·13
6382·11·29
63932·71
64027·5
641 − 660
641641
6422·3·107
643643
64422·7·23
6453·5·43
6462·17·19
647647
64823·34
64911·59
6502·52·13
6513·7·31
65222·163
653653
6542·3·109
6555·131
65624·41
65732·73
6582·7·47
659659
66022·3·5·11
661 − 680
661661
6622·331
6633·13·17
66423·83
6655·7·19
6662·32·37
66723·29
66822·167
6693·223
6702·5·67
67111·61
67225·3·7
673673
6742·337
67533·52
67622·132
677677
6782·3·113
6797·97
68023·5·17
681 − 700
6813·227
6822·11·31
683683
68422·32·19
6855·137
6862·73
6873·229
68824·43
68913·53
6902·3·5·23
691691
69222·173
69332·7·11
6942·347
6955·139
69623·3·29
69717·41
6982·349
6993·233
70022·52·7

701 to 800

701 − 720
701701
7022·33·13
70319·37
70426·11
7053·5·47
7062·353
7077·101
70822·3·59
709709
7102·5·71
71132·79
71223·89
71323·31
7142·3·7·17
7155·11·13
71622·179
7173·239
7182·359
719719
72024·32·5
721 − 740
7217·103
7222·192
7233·241
72422·181
72552·29
7262·3·112
727727
72823·7·13
72936
7302·5·73
73117·43
73222·3·61
733733
7342·367
7353·5·72
73625·23
73711·67
7382·32·41
739739
74022·5·37
741 − 760
7413·13·19
7422·7·53
743743
74423·3·31
7455·149
7462·373
74732·83
74822·11·17
7497·107
7502·3·53
751751
75224·47
7533·251
7542·13·29
7555·151
75622·33·7
757757
7582·379
7593·11·23
76023·5·19
761 − 780
761761
7622·3·127
7637·109
76422·191
76532·5·17
7662·383
76713·59
76828·3
769769
7702·5·7·11
7713·257
77222·193
773773
7742·32·43
77552·31
77623·97
7773·7·37
7782·389
77919·41
78022·3·5·13
781 − 800
78111·71
7822·17·23
78333·29
78424·72
7855·157
7862·3·131
787787
78822·197
7893·263
7902·5·79
7917·113
79223·32·11
79313·61
7942·397
7953·5·53
79622·199
797797
7982·3·7·19
79917·47
80025·52

801 to 900

801 - 820
80132·89
8022·401
80311·73
80422·3·67
8055·7·23
8062·13·31
8073·269
80823·101
809809
8102·34·5
811811
81222·7·29
8133·271
8142·11·37
8155·163
81624·3·17
81719·43
8182·409
81932·7·13
82022·5·41
821 - 840
821821
8222·3·137
823823
82423·103
8253·52·11
8262·7·59
827827
82822·32·23
829829
8302·5·83
8313·277
83226·13
83372·17
8342·3·139
8355·167
83622·11·19
83733·31
8382·419
839839
84023·3·5·7
841 - 860
841292
8422·421
8433·281
84422·211
8455·132
8462·32·47
8477·112
84824·53
8493·283
8502·52·17
85123·37
85222·3·71
853853
8542·7·61
85532·5·19
85623·107
857857
8582·3·11·13
859859
86022·5·43
861 - 880
8613·7·41
8622·431
863863
86425·33
8655·173
8662·433
8673·172
86822·7·31
86911·79
8702·3·5·29
87113·67
87223·109
87332·97
8742·19·23
87553·7
87622·3·73
877877
8782·439
8793·293
88024·5·11
881 - 900
881881
8822·32·72
883883
88422·13·17
8853·5·59
8862·443
887887
88823·3·37
8897·127
8902·5·89
89134·11
89222·223
89319·47
8942·3·149
8955·179
89627·7
8973·13·23
8982·449
89929·31
90022·32·52

901 to 1000

901 - 920
90117·53
9022·11·41
9033·7·43
90423·113
9055·181
9062·3·151
907907
90822·227
90932·101
9102·5·7·13
911911
91224·3·19
91311·83
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919919
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921 - 940
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929929
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9355·11·17
93623·32·13
937937
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941 - 960
941941
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961 - 980
961312
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967967
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971971
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9753·52·13
97624·61
977977
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97911·89
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981 - 1000
98132·109
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983983
98423·3·41
9855·197
9862·17·29
9873·7·47
98822·13·19
98923·43
9902·32·5·11
991991
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9942·7·71
9955·199
99622·3·83
997997
9982·499
99933·37
100023·53
gollark: Why not just let people manually define that when "marrying"?
gollark: Thus, break marriage into its essential components and allow them to be manually done separately with fewer restrictions.
gollark: Exactly, If it's a recognized thing by the government they can apply annoying constraints, like they did with gay marriage before.
gollark: Wait, I have an even BETTER idea: HYPERGRAPHS.
gollark: It's not based because you can't marry unilaterally.

See also

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