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
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
821 - 840
821 821
822 2·3·137
823 823
824 23·103
825 3·52·11
826 2·7·59
827 827
828 22·32·23
829 829
830 2·5·83
831 3·277
832 26·13
833 72·17
834 2·3·139
835 5·167
836 22·11·19
837 33·31
838 2·419
839 839
840 23·3·5·7
841 - 860
841 292
842 2·421
843 3·281
844 22·211
845 5·132
846 2·32·47
847 7·112
848 24·53
849 3·283
850 2·52·17
851 23·37
852 22·3·71
853 853
854 2·7·61
855 32·5·19
856 23·107
857 857
858 2·3·11·13
859 859
860 22·5·43
861 - 880
861 3·7·41
862 2·431
863 863
864 25·33
865 5·173
866 2·433
867 3·172
868 22·7·31
869 11·79
870 2·3·5·29
871 13·67
872 23·109
873 32·97
874 2·19·23
875 53·7
876 22·3·73
877 877
878 2·439
879 3·293
880 24·5·11
881 - 900
881 881
882 2·32·72
883 883
884 22·13·17
885 3·5·59
886 2·443
887 887
888 23·3·37
889 7·127
890 2·5·89
891 34·11
892 22·223
893 19·47
894 2·3·149
895 5·179
896 27·7
897 3·13·23
898 2·449
899 29·31
90022·32·52

901 to 1000

901 - 920
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
921 - 940
921 3·307
922 2·461
923 13·71
924 22·3·7·11
925 52·37
926 2·463
927 32·103
928 25·29
929 929
930 2·3·5·31
931 72·19
932 22·233
933 3·311
934 2·467
935 5·11·17
936 23·32·13
937 937
938 2·7·67
939 3·313
940 22·5·47
941 - 960
941 941
942 2·3·157
943 23·41
944 24·59
945 33·5·7
946 2·11·43
947 947
948 22·3·79
949 13·73
950 2·52·19
951 3·317
952 23·7·17
953 953
954 2·32·53
955 5·191
956 22·239
957 3·11·29
958 2·479
959 7·137
960 26·3·5
961 - 980
961 312
962 2·13·37
963 32·107
964 22·241
965 5·193
966 2·3·7·23
967 967
968 23·112
969 3·17·19
970 2·5·97
971 971
972 22·35
973 7·139
974 2·487
975 3·52·13
976 24·61
977 977
978 2·3·163
979 11·89
980 22·5·72
981 - 1000
981 32·109
982 2·491
983 983
984 23·3·41
985 5·197
986 2·17·29
987 3·7·47
988 22·13·19
989 23·43
990 2·32·5·11
991 991
992 25·31
993 3·331
994 2·7·71
995 5·199
996 22·3·83
997 997
998 2·499
999 33·37
1000 23·53

See also

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