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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.

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 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 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
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

See also

table, prime, factors, tables, contain, prime, factorization, natural, numbers, from, 1000, when, prime, number, prime, factorization, just, itself, written, bold, below, number, called, unit, prime, factors, neither, prime, composite, contents, properties, 10. 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 Contents 1 Properties 2 1 to 100 3 101 to 200 4 201 to 300 5 301 to 400 6 401 to 500 7 501 to 600 8 601 to 700 9 701 to 800 10 801 to 900 11 901 to 1000 12 See alsoPropertiesMany 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 W 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 W 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 W n gt 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 W 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 W n k so it is composite if k gt 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 gt 1 for all multiplicities it is of the form am for some a gt 1 and m gt 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 l n is 1 if W n is even and is 1 if W n is odd The Mobius function m n is 0 if n is not square free Otherwise m n is 1 if W n is even and is 1 if W n is odd A sphenic number has W 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 gt 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 1001 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 52101 to 200101 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 52201 to 300201 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 52301 to 400301 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 52401 to 500401 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 53501 to 600501 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 52601 to 700601 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 7701 to 800701 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 52801 to 900801 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 52901 to 1000901 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 53See alsoFundamental 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 Retrieved from https en wikipedia org w index php title Table of prime factors amp oldid 1218851008, wikipedia, wiki, book, books, library,

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