Table of prime factors

Template:Pp-sock Template:Short description 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. Template:More citations needed

Properties

Many properties of a natural number <math>n</math> can be seen or directly computed from the prime factorization of <math>n</math>.

• The multiplicity of a prime factor <math>p</math> of <math>n</math> is the largest exponent <math>m</math> for which <math>p^m</math> divides <math>n</math>. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is <math>1</math> (since <math>p = p^1</math>). The multiplicity of a prime which does not divide <math>n</math> may be called <math>0</math> or may be considered undefined.
• <math>\omega(n)</math> and <math>\Omega(n)</math>, the prime omega functions, count the number of prime factors of a natural number <math>n</math>.
  • <math>\omega(n)</math> (little omega) is the number of distinct prime factors of <math>n</math>.
  • <math>\Omega(n)</math> (big omega) is the number of prime factors of <math>n</math> counted with multiplicity (so it is the sum of all prime factor multiplicities).
• A prime number has <math>\Omega(n) = \omega(n) = 1</math>. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 Template:OEIS. There are many special types of prime numbers.
• A composite number has <math>\Omega(n) \ge \omega(n) > 1</math>. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 Template:OEIS. All numbers above 1 are either prime or composite. 1 is neither.
• A semiprime has <math>\Omega(n) = 2</math> (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 Template:OEIS.
• A <math>k</math>-almost prime (for a natural number <math>k</math>) has <math>\Omega(n) = k</math> (so it is composite if <math>k > 1</math>).
• An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 Template: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 Template:OEIS. All integers are either even or odd.
• A square has even multiplicity for all prime factors (it is of the form <math>a^2</math> for some <math>a</math>). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 Template:OEIS.
• A cube has all multiplicities divisible by 3 (it is of the form <math>a^3</math> for some <math>a</math>). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 Template:OEIS.
• A perfect power has a common divisor <math>m > 1</math> for all multiplicities (it is of the form <math>a^m</math> for some <math>a > 1</math> and <math>m > 1</math>). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 Template:OEIS. 1 is sometimes included.
• A powerful number (also called squarefull) has multiplicity greater than 1 for all its prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 Template:OEIS.
• A prime power has only one prime factor, i.e. <math>\omega(n) = 1</math>. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 Template: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 Template:OEIS.
• A square-free integer has no prime factor with multiplicity greater than 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 Template:OEIS. A number where some but not all prime factors have multiplicity greater than 1 is neither square-free nor squarefull, but squareful.
• The Liouville function <math>\lambda(n)</math> is 1 if <math>\Omega(n)</math> is even, and is -1 if <math>\Omega(n)</math> is odd.
• The Möbius function <math>\mu(n)</math> is 0 if <math>n</math> is not square-free. Otherwise <math>\mu(n)</math> is 1 if <math>\Omega(n)</math> is even, and is −1 if <math>\Omega(n)</math> is odd.
• A sphenic number is square-free and the product of 3 distinct primes, i.e. it has <math>\omega(n) = \Omega(n) = 3</math>. The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 Template:OEIS.
• <math>a_0(n)</math>, sometimes called the integer logarithm, is the sum of primes dividing <math>n</math>, counted with multiplicity. It is an additive function.
• A Ruth-Aaron pair is a pair of two consecutive numbers <math>(n, n+1)</math> with <math>a_0(n) = a_0(n+1)</math>. The first (by <math>n</math> value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 Template:OEIS. Another definition is where the same prime is only counted once; if so, the first (by <math>n</math> value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 Template:OEIS.
• A primorial <math>p_n\#</math> is the product of all primes from 2 to <math>p_n</math>. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 Template:OEIS. <math>1\# = 1</math> is sometimes included.
• A factorial <math>n!</math> is the product of all numbers from 1 to <math>n</math>. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 Template:OEIS. <math>0! = 1</math> is sometimes included.
• A <math>k</math>-smooth number (for a natural number <math>k</math>) has its prime factors <math>\le k</math> (so it is also <math>j</math>-smooth for any <math>j > k</math>).
• <math>m</math> is smoother than <math>n</math> if the largest prime factor of <math>m</math> is less than the largest of <math>n</math>.
• A regular number has no prime factor greater than 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 Template:OEIS.
• A <math>k</math>-powersmooth number has all <math>p^m \le k</math> where <math>p</math> is a prime factor with multiplicity <math>m</math>.
• 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 Template: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 Template: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 Template:OEIS.
• An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
• <math>gcd(m, n)</math> (greatest common divisor of <math>m</math> and <math>n</math>) is the product of all prime factors which are both in <math>m</math> and <math>n</math> (with the smallest multiplicity for <math>m</math> and <math>n</math>).
• <math>m</math> and <math>n</math> are coprime (also called relatively prime) if they have no common prime factors, which implies <math>gcd(m, n) = 1</math>.
• <math>lcm(m, n)</math> (least common multiple of <math>m</math> and <math>n</math>) is the product of all prime factors of <math>m</math> or <math>n</math> (with the largest multiplicity for <math>m</math> or <math>n</math>).
• <math>gcd(m, n) \times lcm(m, n) = m \times n</math>. Finding the prime factors is often harder than computing <math>gcd</math> and <math>lcm</math> using other algorithms which do not require known prime factorization.
• <math>m</math> is a divisor of <math>n</math> (also called <math>m</math> divides <math>n</math>, or <math>n</math> is divisible by <math>m</math>) if all prime factors of <math>m</math> have at least the same multiplicity in <math>n</math>.
• The divisors of <math>n</math> are all products of some or all prime factors of <math>n</math> (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

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