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 p^m 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 a^m 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 p^m ≤ 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 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 to compute gcd and lcm with 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
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

21 − 40
21	3·7
22	2·11
23	23
24	2³·3
25	5²
26	2·13
27	3³
28	2²·7
29	29
30	2·3·5
31	31
32	2⁵
33	3·11
34	2·17
35	5·7
36	2²·3²
37	37
38	2·19
39	3·13
40	2³·5

41 − 60
41	41
42	2·3·7
43	43
44	2²·11
45	3²·5
46	2·23
47	47
48	2⁴·3
49	7²
50	2·5²
51	3·17
52	2²·13
53	53
54	2·3³
55	5·11
56	2³·7
57	3·19
58	2·29
59	59
60	2²·3·5

61 − 80
61	61
62	2·31
63	3²·7
64	2⁶
65	5·13
66	2·3·11
67	67
68	2²·17
69	3·23
70	2·5·7
71	71
72	2³·3²
73	73
74	2·37
75	3·5²
76	2²·19
77	7·11
78	2·3·13
79	79
80	2⁴·5

81 − 100
81	3⁴
82	2·41
83	83
84	2²·3·7
85	5·17
86	2·43
87	3·29
88	2³·11
89	89
90	2·3²·5
91	7·13
92	2²·23
93	3·31
94	2·47
95	5·19
96	2⁵·3
97	97
98	2·7²
99	3²·11
100	2²·5²

101 to 200

101 − 120
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

121 − 140
121	11²
122	2·61
123	3·41
124	2²·31
125	5³
126	2·3²·7
127	127
128	2⁷
129	3·43
130	2·5·13
131	131
132	2²·3·11
133	7·19
134	2·67
135	3³·5
136	2³·17
137	137
138	2·3·23
139	139
140	2²·5·7

141 − 160
141	3·47
142	2·71
143	11·13
144	2⁴·3²
145	5·29
146	2·73
147	3·7²
148	2²·37
149	149
150	2·3·5²
151	151
152	2³·19
153	3²·17
154	2·7·11
155	5·31
156	2²·3·13
157	157
158	2·79
159	3·53
160	2⁵·5

161 − 180
161	7·23
162	2·3⁴
163	163
164	2²·41
165	3·5·11
166	2·83
167	167
168	2³·3·7
169	13²
170	2·5·17
171	3²·19
172	2²·43
173	173
174	2·3·29
175	5²·7
176	2⁴·11
177	3·59
178	2·89
179	179
180	2²·3²·5

181 − 200
181	181
182	2·7·13
183	3·61
184	2³·23
185	5·37
186	2·3·31
187	11·17
188	2²·47
189	3³·7
190	2·5·19
191	191
192	2⁶·3
193	193
194	2·97
195	3·5·13
196	2²·7²
197	197
198	2·3²·11
199	199
200	2³·5²

201 to 300

201 − 220
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

221 − 240
221	13·17
222	2·3·37
223	223
224	2⁵·7
225	3²·5²
226	2·113
227	227
228	2²·3·19
229	229
230	2·5·23
231	3·7·11
232	2³·29
233	233
234	2·3²·13
235	5·47
236	2²·59
237	3·79
238	2·7·17
239	239
240	2⁴·3·5

241 − 260
241	241
242	2·11²
243	3⁵
244	2²·61
245	5·7²
246	2·3·41
247	13·19
248	2³·31
249	3·83
250	2·5³
251	251
252	2²·3²·7
253	11·23
254	2·127
255	3·5·17
256	2⁸
257	257
258	2·3·43
259	7·37
260	2²·5·13

261 − 280
261	3²·29
262	2·131
263	263
264	2³·3·11
265	5·53
266	2·7·19
267	3·89
268	2²·67
269	269
270	2·3³·5
271	271
272	2⁴·17
273	3·7·13
274	2·137
275	5²·11
276	2²·3·23
277	277
278	2·139
279	3²·31
280	2³·5·7

281 − 300
281	281
282	2·3·47
283	283
284	2²·71
285	3·5·19
286	2·11·13
287	7·41
288	2⁵·3²
289	17²
290	2·5·29
291	3·97
292	2²·73
293	293
294	2·3·7²
295	5·59
296	2³·37
297	3³·11
298	2·149
299	13·23
300	2²·3·5²

301 to 400

301 − 320
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

321 − 340
321	3·107
322	2·7·23
323	17·19
324	2²·3⁴
325	5²·13
326	2·163
327	3·109
328	2³·41
329	7·47
330	2·3·5·11
331	331
332	2²·83
333	3²·37
334	2·167
335	5·67
336	2⁴·3·7
337	337
338	2·13²
339	3·113
340	2²·5·17

341 − 360
341	11·31
342	2·3²·19
343	7³
344	2³·43
345	3·5·23
346	2·173
347	347
348	2²·3·29
349	349
350	2·5²·7
351	3³·13
352	2⁵·11
353	353
354	2·3·59
355	5·71
356	2²·89
357	3·7·17
358	2·179
359	359
360	2³·3²·5

361 − 380
361	19²
362	2·181
363	3·11²
364	2²·7·13
365	5·73
366	2·3·61
367	367
368	2⁴·23
369	3²·41
370	2·5·37
371	7·53
372	2²·3·31
373	373
374	2·11·17
375	3·5³
376	2³·47
377	13·29
378	2·3³·7
379	379
380	2²·5·19

381 − 400
381	3·127
382	2·191
383	383
384	2⁷·3
385	5·7·11
386	2·193
387	3²·43
388	2²·97
389	389
390	2·3·5·13
391	17·23
392	2³·7²
393	3·131
394	2·197
395	5·79
396	2²·3²·11
397	397
398	2·199
399	3·7·19
400	2⁴·5²

401 to 500

401 − 420
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

421 − 440
421	421
422	2·211
423	3²·47
424	2³·53
425	5²·17
426	2·3·71
427	7·61
428	2²·107
429	3·11·13
430	2·5·43
431	431
432	2⁴·3³
433	433
434	2·7·31
435	3·5·29
436	2²·109
437	19·23
438	2·3·73
439	439
440	2³·5·11

441 − 460
441	3²·7²
442	2·13·17
443	443
444	2²·3·37
445	5·89
446	2·223
447	3·149
448	2⁶·7
449	449
450	2·3²·5²
451	11·41
452	2²·113
453	3·151
454	2·227
455	5·7·13
456	2³·3·19
457	457
458	2·229
459	3³·17
460	2²·5·23

461 − 480
461	461
462	2·3·7·11
463	463
464	2⁴·29
465	3·5·31
466	2·233
467	467
468	2²·3²·13
469	7·67
470	2·5·47
471	3·157
472	2³·59
473	11·43
474	2·3·79
475	5²·19
476	2²·7·17
477	3²·53
478	2·239
479	479
480	2⁵·3·5

481 − 500
481	13·37
482	2·241
483	3·7·23
484	2²·11²
485	5·97
486	2·3⁵
487	487
488	2³·61
489	3·163
490	2·5·7²
491	491
492	2²·3·41
493	17·29
494	2·13·19
495	3²·5·11
496	2⁴·31
497	7·71
498	2·3·83
499	499
500	2²·5³

501 to 600

501 − 520
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

521 − 540
521	521
522	2·3²·29
523	523
524	2²·131
525	3·5²·7
526	2·263
527	17·31
528	2⁴·3·11
529	23²
530	2·5·53
531	3²·59
532	2²·7·19
533	13·41
534	2·3·89
535	5·107
536	2³·67
537	3·179
538	2·269
539	7²·11
540	2²·3³·5

541 − 560
541	541
542	2·271
543	3·181
544	2⁵·17
545	5·109
546	2·3·7·13
547	547
548	2²·137
549	3²·61
550	2·5²·11
551	19·29
552	2³·3·23
553	7·79
554	2·277
555	3·5·37
556	2²·139
557	557
558	2·3²·31
559	13·43
560	2⁴·5·7

561 − 580
561	3·11·17
562	2·281
563	563
564	2²·3·47
565	5·113
566	2·283
567	3⁴·7
568	2³·71
569	569
570	2·3·5·19
571	571
572	2²·11·13
573	3·191
574	2·7·41
575	5²·23
576	2⁶·3²
577	577
578	2·17²
579	3·193
580	2²·5·29

581 − 600
581	7·83
582	2·3·97
583	11·53
584	2³·73
585	3²·5·13
586	2·293
587	587
588	2²·3·7²
589	19·31
590	2·5·59
591	3·197
592	2⁴·37
593	593
594	2·3³·11
595	5·7·17
596	2²·149
597	3·199
598	2·13·23
599	599
600	2³·3·5²

601 to 700

601 − 620
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

621 − 640
621	3³·23
622	2·311
623	7·89
624	2⁴·3·13
625	5⁴
626	2·313
627	3·11·19
628	2²·157
629	17·37
630	2·3²·5·7
631	631
632	2³·79
633	3·211
634	2·317
635	5·127
636	2²·3·53
637	7²·13
638	2·11·29
639	3²·71
640	2⁷·5

641 − 660
641	641
642	2·3·107
643	643
644	2²·7·23
645	3·5·43
646	2·17·19
647	647
648	2³·3⁴
649	11·59
650	2·5²·13
651	3·7·31
652	2²·163
653	653
654	2·3·109
655	5·131
656	2⁴·41
657	3²·73
658	2·7·47
659	659
660	2²·3·5·11

661 − 680
661	661
662	2·331
663	3·13·17
664	2³·83
665	5·7·19
666	2·3²·37
667	23·29
668	2²·167
669	3·223
670	2·5·67
671	11·61
672	2⁵·3·7
673	673
674	2·337
675	3³·5²
676	2²·13²
677	677
678	2·3·113
679	7·97
680	2³·5·17

681 − 700
681	3·227
682	2·11·31
683	683
684	2²·3²·19
685	5·137
686	2·7³
687	3·229
688	2⁴·43
689	13·53
690	2·3·5·23
691	691
692	2²·173
693	3²·7·11
694	2·347
695	5·139
696	2³·3·29
697	17·41
698	2·349
699	3·233
700	2²·5²·7

701 to 800

701 − 720
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

721 − 740
721	7·103
722	2·19²
723	3·241
724	2²·181
725	5²·29
726	2·3·11²
727	727
728	2³·7·13
729	3⁶
730	2·5·73
731	17·43
732	2²·3·61
733	733
734	2·367
735	3·5·7²
736	2⁵·23
737	11·67
738	2·3²·41
739	739
740	2²·5·37

741 − 760
741	3·13·19
742	2·7·53
743	743
744	2³·3·31
745	5·149
746	2·373
747	3²·83
748	2²·11·17
749	7·107
750	2·3·5³
751	751
752	2⁴·47
753	3·251
754	2·13·29
755	5·151
756	2²·3³·7
757	757
758	2·379
759	3·11·23
760	2³·5·19

761 − 780
761	761
762	2·3·127
763	7·109
764	2²·191
765	3²·5·17
766	2·383
767	13·59
768	2⁸·3
769	769
770	2·5·7·11
771	3·257
772	2²·193
773	773
774	2·3²·43
775	5²·31
776	2³·97
777	3·7·37
778	2·389
779	19·41
780	2²·3·5·13

781 − 800
781	11·71
782	2·17·23
783	3³·29
784	2⁴·7²
785	5·157
786	2·3·131
787	787
788	2²·197
789	3·263
790	2·5·79
791	7·113
792	2³·3²·11
793	13·61
794	2·397
795	3·5·53
796	2²·199
797	797
798	2·3·7·19
799	17·47
800	2⁵·5²

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
900	2²·3²·5²

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