168 (number)

Natural number

| <- 167  | 168 |  169 ->

|
|

← 160 161 162 163 164 165 166 167 168 169 →

* List of numbers
* Integers

← 0 100 200 300 400 500 600 700 800 900 →

Cardinal| one hundred sixty-eight
Ordinal| 168th
(one hundred sixty-eighth)
Factorization| 23 × 3 × 7
Divisors| 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
Greek numeral| ΡΞΗ´
Roman numeral| CLXVIII, clxviii
Binary| 101010002
Ternary| 200203
Senary| 4406
Octal| 2508
Duodecimal| 12012
Hexadecimal| A816

168 (one hundred [and] sixty-eight) is the natural number following 167 and preceding 169.

It is the number of hours in a week, or 7×24 hours. 168 is the fourth Dedekind number,[1] and the 128th composite number.

## Mathematics

[edit]

### Number theory

[edit]

168 is the fourth Dedekind number,[1] and one of sixty-five idoneal numbers.[2] It is one less than a square (132), equal to the product of the first two perfect numbers[3]

168 = 6 × 28. {\displaystyle 168=6\times 28.}

There are 168 primes less than 1000.[a]

#### Composite index

[edit]

The 128th composite number is 168,[4] one of a few numbers  n {\displaystyle n} in the list of composites whose indices are the product of strings of digits of  n {\displaystyle n} in decimal representation.

The first nine  n {\displaystyle n} with this property are the following:[4]

* 48 as the 32nd (4 × 8 = 32) [b]
* 68 as the 48th (6 × 8 = 48)
* 78 is the 56th (7 × 8 = 56)
* 88 as the 64th (82 = 64)
* 98 as the 72nd (9 × 8 = 72)
* 128 as the 96th (12 × 8 = 96)
* 138 as the 104th (13 × 8 = 104)
* 158 as the 120th (15 × 8 = 120)
* 168 as the 128th (16 × 8 = 128) [c]

The next such number is 198 where 19 × 8 = 152. The median between twenty-one integers [48, 68] is 58, where 148 is the median of forty-one integers [168, 128].

#### Totient and sigma values

[edit]

For the Euler totient φ ( n ) {\displaystyle \varphi (n)} there is  φ ( 168 ) = 48 {\displaystyle \varphi (168)=48} ,[5] where  φ ( 48 ) = 16 {\displaystyle \varphi (48)=16} is also equivalent to the number of divisors of 168;[6] only eleven numbers have a totient of 48:{65, 104, 105, 112, 130, 140, 144, 156, 168, 180, 210}.[5][d]

The number of divisors of 168 is 16,[8] making it a largely composite number.[9]

408,[e] with a different permutation of the digits {0, 4, 8} where 048 is 48, has an totient of 128. So does the sum-of-divisors of 168,[11]

σ ( 168 ) = 480 = 2 5 × 3 × 5 {\displaystyle \sigma (168)=480=2^{5}\times 3\times 5}

as one of nine numbers total to have a totient of 128.[5]

48 sets the sixteenth record for sum-of-divisors of positive integers (of 124), and the seventeenth record value is 168,[12] from six numbers (60, 78, 92, 123, 143, and 167).[11]

The difference between 168 and 48 is the factorial of five (120), where their sum is the cube of six (216).

#### Idoneal number

[edit]

Leonhard Euler noted 65 idoneal numbers (the most known, of only a maximum possible of two more), such that  x 2 + D y 2 {\displaystyle x^{2}+Dy^{2}} for an integer  D {\displaystyle D} , expressible in only one way, yields a prime power or twice a prime power.[2][13]

Of these, 168 is the forty-fourth, where the smallest number to not be idoneal is the fifth prime number 11.[2] The largest such number 1848 (that is equivalent with the number of edges in the union of two cycle graphs of order 42)[14] contains a total of thirty-two divisors whose arithmetic mean is 180[15][16] (the second-largest number to have a totient of 48).[5] Preceding 1848 in the list of idoneal numbers is 1365,[f] whose arithmetic mean of divisors is equal to 168[15][16] (while 1365 has a totient of 576 = 242).

Where 48 is the 27th ideoneal number, 408 is the 58th.[2][g] On the other hand, the total count of known idoneal numbers (65), that is also equal to the sum of ten integers [2, ..., 11], has a sum-of-divisors of 84 (or, one-half of 168).[11]

#### Numbers of the form 2n

[edit]

In base 10, 168 is the largest of ninety-two known  n {\displaystyle n} such that  2 n {\displaystyle 2^{n}} does not contain all numerical digits from that base (i.e. 0, 1, 2, ..., 9).[18]

2 68 {\displaystyle 2^{68}} is the first number to have such an expression where between the next two  n {\displaystyle n} is an interval of ten integers: [70, 79];[18] the median values between these are (75, 74), where the smaller of these two values represents the composite index of 100.[4][h]

#### Cunningham number

[edit]

As a number of the form  b n ± 1 {\displaystyle b^{n}\pm 1} for positive integers  n {\displaystyle n} ,  b {\displaystyle b} and  b {\displaystyle b} not a perfect power, 168 is the thirty-second Cunningham number,[22] where it is one less than a square:

168 = 13 2 − 1. {\displaystyle 168=13^{2}-1.}

On the other hand, 168 is one more than the third member of the fourth chain of nearly doubled primes of the first kind {41, 83, 167},[23][24] where 167 represents the thirty-ninth prime[25] (with 39 × 2 = 78). The smallest such chain is {2, 5, 11, 23, 47}.

#### Eisenstein series

[edit]

168 is also coefficient four in the expansion of Eisenstein series E 2 {\displaystyle E_{2}} ,[26] which also includes 144 and 96 (or 48 × 2) as the fifth and third coefficients, respectively — these have a sum of 240, which follows 144 and 187 in the list of successive composites  A ( n + 1 ) = A ( n ) {\displaystyle A(n+1)=A(n)} ;cf.[4] the latter holds a sum-of-divisors of 216 = 63,[11] which is the 168th composite number.[4]

### Abstract algebra

[edit]

168 is the number of maximal chains in the Bruhat order of symmetric group S 4 , {\displaystyle \mathrm {S} _{4},} [27] which is the largest solvable symmetric group with a total of  4 ! = 24 {\displaystyle 4!=24} elements.

168 is the order of the second smallest nonabelian simple group P S L ( 2 , 7 ) . {\displaystyle \mathrm {PSL} (2,7).} From Hurwitz's automorphisms theorem, 168 is the maximum possible number of automorphisms of a genus 3 Riemann surface, this maximum being achieved by the Klein quartic, whose symmetry group is  P S L ( 2 , 7 ) {\displaystyle \mathrm {PSL} (2,7)} ;[28] the Fano plane, isomorphic to the Klein group, has 168 symmetries.

## In other fields

[edit]

### Numerology

[edit]

Some Chinese consider 168 a lucky number, because 一六八 ("168") Mandarin pinyin: yīliùbā is roughly homophonous with the phrase "一路發" Mandarin pinyin: yīlùfā which means "fortune all the way", or, as the United States Mint claims, "Prosperity Forever".[29]

## Notes

[edit]

1. ^ (168, 1000) un-inclusive corresponds to a range of 831 integers, which is a value in equivalence with the composite index of 1000 = 103.[4]
2. ^ 32 is the twentieth composite.
3. ^ 128 = 64 × 2 = 32 × 4, with 96 = 48 × 2, where also 16810 = 12012 (in duodecimal).
On the other hand, 28 is the 18th composite number,[4]
4. ^ The latter (210) is the 20th triangle number.[7]
5. ^ 505, which is the magic constant of a  10 × 10 {\displaystyle 10\times 10} magic square,[10] is the 408th composite number.
6. ^ 1365 ÷ 3 = 455 is the sum of (the first) ten terms in the sequence of numbers k ∈ {1, 2, 3, 4, 7, 8, 16, 31, 127, 256} such that k and k \+ 1 are prime powers.[17]
7. ^ 840, with thirty-two divisors (the number with the largest number of divisors less than 1000), is the fourth-largest idoneal number. 88, 78, 58, 28, and 18 are also idoneal numbers, including 210 and 105 (numbers with totients of 48).[2]
8. ^ In the iterative list of the A(n)-th composite number with A(1) = 11 where A(n + 1) = A(n), the first few elements are
11, 20, 32, 48, 68, 93, 124, ...[19]
which is preceded at 11 with the analogous list of successive super-primes[20] and primes[21] 11, 5, 3, 2, 1 (if the unit is a zeroth prime).
The sum of these elements 1, 2, 3, 5, 11, 20, 32 is 74, with 32 + 68 = 100, and 48 in between.

## References

[edit]

1. ^ a b Sloane, N. J. A. (ed.). "Sequence A000372 (Dedekind numbers: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-01.
2. ^ a b c d e "Sloane's A000926 : Euler's "numerus idoneus" (or "numeri idonei", or idoneal, or suitable, or convenient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-28.
3. ^ Sloane, N. J. A. (ed.). "Sequence A000396 (Perfect numbers k: k is equal to the sum of the proper divisors of k.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-01.
4. ^ a b c d e f g Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers: numbers n of the form x*y for x greater than 1 and y greater than 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-05.
5. ^ a b c d Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-05.
6. ^ Sloane, N. J. A. (ed.). "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-05.
7. ^ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers: 0 + 1 + 2 + ... + n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
8. ^ Sloane, N. J. A. (ed.). "Sequence A000005 (d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-05.
9. ^ Sloane, N. J. A. (ed.). "Sequence A067128 (Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
10. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) as n*(n^2 + 1)/2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
11. ^ a b c d Sloane, N. J. A. (ed.). "Sequence A000203 (The sum of divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
12. ^ Sloane, N. J. A. (ed.). "Sequence A034885 (Record values of sigma(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-07-06.
13. ^ Euler, Leonard (1806). "Illustratio paradoxi circa progressionem numerorum idoneorum sive congruorum". Nova Acta Academiae Scientarum Imperialis Petropolitinae. 15. Russian Academy of Sciences: 29–32\. arXiv:math/0507352. S2CID 118287274.
14. ^ Sloane, N. J. A. (ed.). "Sequence A005563 (a(n) as n*(n+2) equal to (n+1)^2 - 1.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
15. ^ a b Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
16. ^ a b Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-07-16.
17. ^ Sloane, N. J. A. (ed.). "Sequence A006549 (Numbers k such that k and k+1 are prime powers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-09.
18. ^ a b "Sloane's A130696: Numbers k such that 2^k does not contain all ten decimal digits". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-19.
19. ^ Sloane, N. J. A. (ed.). "Sequence A059407 (a(n+1) as the a(n)-th composite number, with a(1) equal to 11.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
20. ^ Sloane, N. J. A. (ed.). "Sequence A006450 (Prime-indexed primes: primes with prime subscripts.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
21. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-08.
22. ^ Sloane, N. J. A. (ed.). "Sequence A080262 (Cunningham numbers: of the form a^b +- 1, where a, b are greater than or equal to 2.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
23. ^ Sloane, N. J. A. (ed.). "Sequence A005602 (Smallest prime beginning a complete Cunningham chain of length n (of the first kind).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
24. ^ Sloane, N. J. A. (ed.). "Sequence A348855 (a(1) is 1. If a(n) is prime, a(n+1) is 2*a(n) + 1. If a(n) is not prime, a(n+1) is the least prime not already in the sequence.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
25. ^ Sloane, N. J. A. (ed.). "Sequence A000040 (The prime numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-20.
26. ^ Sloane, N. J. A. (ed.). "Sequence A006352 (Coefficients in expansion of Eisenstein series E_2 (also called E_1 or G_2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-02.
27. ^ Sloane, N. J. A. (ed.). "Sequence A061710 (Number of maximal chains in the Bruhat order of S_n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-04-01.
28. ^ "week214". math.ucr.edu. Retrieved 9 April 2023.
29. ^ "$1 Prosperity Forever 168 Note - US Mint". Retrieved 9 April 2023.

## External links

[edit]

Wikimedia Commons has media related to 168 (number).

* Number Facts and Trivia: 168
* The Number 168
* The Positive Integer 168
* Prime curiosities: 168

* v
* t
* e

Integers

* 0
* 1
* 2
* 3
* 4
* 5
* 6
* 7
* 8
* 9

* 10
* 11
* 12
* 13
* 14
* 15
* 16
* 17
* 18
* 19

* 20
* 21
* 22
* 23
* 24
* 25
* 26
* 27
* 28
* 29

* 30
* 31
* 32
* 33
* 34
* 35
* 36
* 37
* 38
* 39

* 40
* 41
* 42
* 43
* 44
* 45
* 46
* 47
* 48
* 49

* 50
* 51
* 52
* 53
* 54
* 55
* 56
* 57
* 58
* 59

* 60
* 61
* 62
* 63
* 64
* 65
* 66
* 67
* 68
* 69

* 70
* 71
* 72
* 73
* 74
* 75
* 76
* 77
* 78
* 79

* 80
* 81
* 82
* 83
* 84
* 85
* 86
* 87
* 88
* 89

* 90
* 91
* 92
* 93
* 94
* 95
* 96
* 97
* 98
* 99

100s

* 100
* 101
* 102
* 103
* 104
* 105
* 106
* 107
* 108
* 109

* 110
* 111
* 112
* 113
* 114
* 115
* 116
* 117
* 118
* 119

* 120
* 121
* 122
* 123
* 124
* 125
* 126
* 127
* 128
* 129

* 130
* 131
* 132
* 133
* 134
* 135
* 136
* 137
* 138
* 139

* 140
* 141
* 142
* 143
* 144
* 145
* 146
* 147
* 148
* 149

* 150
* 151
* 152
* 153
* 154
* 155
* 156
* 157
* 158
* 159

* 160
* 161
* 162
* 163
* 164
* 165
* 166
* 167
* 168
* 169

* 170
* 171
* 172
* 173
* 174
* 175
* 176
* 177
* 178
* 179

* 180
* 181
* 182
* 183
* 184
* 185
* 186
* 187
* 188
* 189

* 190
* 191
* 192
* 193
* 194
* 195
* 196
* 197
* 198
* 199

200s

* 200
* 201
* 202
* 203
* 204
* 205
* 206
* 207
* 208
* 209

* 210
* 211
* 212
* 213
* 214
* 215
* 216
* 217
* 218
* 219

* 220
* 221
* 222
* 223
* 224
* 225
* 226
* 227
* 228
* 229

* 230
* 231
* 232
* 233
* 234
* 235
* 236
* 237
* 238
* 239

* 240
* 241
* 242
* 243
* 244
* 245
* 246
* 247
* 248
* 249

* 250
* 251
* 252
* 253
* 254
* 255
* 256
* 257
* 258
* 259

* 260
* 261
* 262
* 263
* 264
* 265
* 266
* 267
* 268
* 269

* 270
* 271
* 272
* 273
* 274
* 275
* 276
* 277
* 278
* 279

* 280
* 281
* 282
* 283
* 284
* 285
* 286
* 287
* 288
* 289

* 290
* 291
* 292
* 293
* 294
* 295
* 296
* 297
* 298
* 299

300s

* 300
* 301
* 302
* 303
* 304
* 305
* 306
* 307
* 308
* 309

* 310
* 311
* 312
* 313
* 314
* 315
* 316
* 317
* 318
* 319

* 320
* 321
* 322
* 323
* 324
* 325
* 326
* 327
* 328
* 329

* 330
* 331
* 332
* 333
* 334
* 335
* 336
* 337
* 338
* 339

* 340
* 341
* 342
* 343
* 344
* 345
* 346
* 347
* 348
* 349

* 350
* 351
* 352
* 353
* 354
* 355
* 356
* 357
* 358
* 359

* 360
* 361
* 362
* 363
* 364
* 365
* 366
* 367
* 368
* 369

* 370
* 371
* 372
* 373
* 374
* 375
* 376
* 377
* 378
* 379

* 380
* 381
* 382
* 383
* 384
* 385
* 386
* 387
* 388
* 389

* 390
* 391
* 392
* 393
* 394
* 395
* 396
* 397
* 398
* 399

400s

* 400
* 401
* 402
* 403
* 404
* 405
* 406
* 407
* 408
* 409

* 410
* 411
* 412
* 413
* 414
* 415
* 416
* 417
* 418
* 419

* 420
* 421
* 422
* 423
* 424
* 425
* 426
* 427
* 428
* 429

* 430
* 431
* 432
* 433
* 434
* 435
* 436
* 437
* 438
* 439

* 440
* 441
* 442
* 443
* 444
* 445
* 446
* 447
* 448
* 449

* 450
* 451
* 452
* 453
* 454
* 455
* 456
* 457
* 458
* 459

* 460
* 461
* 462
* 463
* 464
* 465
* 466
* 467
* 468
* 469

* 470
* 471
* 472
* 473
* 474
* 475
* 476
* 477
* 478
* 479

* 480
* 481
* 482
* 483
* 484
* 485
* 486
* 487
* 488
* 489

* 490
* 491
* 492
* 493
* 494
* 495
* 496
* 497
* 498
* 499

500s

* 500
* 501
* 502
* 503
* 504
* 505
* 506
* 507
* 508
* 509

* 510
* 511
* 512
* 513
* 514
* 515
* 516
* 517
* 518
* 519

* 520
* 521
* 522
* 523
* 524
* 525
* 526
* 527
* 528
* 529

* 530
* 531
* 532
* 533
* 534
* 535
* 536
* 537
* 538
* 539

* 540
* 541
* 542
* 543
* 544
* 545
* 546
* 547
* 548
* 549

* 550
* 551
* 552
* 553
* 554
* 555
* 556
* 557
* 558
* 559

* 560
* 561
* 562
* 563
* 564
* 565
* 566
* 567
* 568
* 569

* 570
* 571
* 572
* 573
* 574
* 575
* 576
* 577
* 578
* 579

* 580
* 581
* 582
* 583
* 584
* 585
* 586
* 587
* 588
* 589

* 590
* 591
* 592
* 593
* 594
* 595
* 596
* 597
* 598
* 599

600s

* 600
* 601
* 602
* 603
* 604
* 605
* 606
* 607
* 608
* 609

* 610
* 611
* 612
* 613
* 614
* 615
* 616
* 617
* 618
* 619

* 620
* 621
* 622
* 623
* 624
* 625
* 626
* 627
* 628
* 629

* 630
* 631
* 632
* 633
* 634
* 635
* 636
* 637
* 638
* 639

* 640
* 641
* 642
* 643
* 644
* 645
* 646
* 647
* 648
* 649

* 650
* 651
* 652
* 653
* 654
* 655
* 656
* 657
* 658
* 659

* 660
* 661
* 662
* 663
* 664
* 665
* 666
* 667
* 668
* 669

* 670
* 671
* 672
* 673
* 674
* 675
* 676
* 677
* 678
* 679

* 680
* 681
* 682
* 683
* 684
* 685
* 686
* 687
* 688
* 689

* 690
* 691
* 692
* 693
* 694
* 695
* 696
* 697
* 698
* 699

700s

* 700
* 701
* 702
* 703
* 704
* 705
* 706
* 707
* 708
* 709

* 710
* 711
* 712
* 713
* 714
* 715
* 716
* 717
* 718
* 719

* 720
* 721
* 722
* 723
* 724
* 725
* 726
* 727
* 728
* 729

* 730
* 731
* 732
* 733
* 734
* 735
* 736
* 737
* 738
* 739

* 740
* 741
* 742
* 743
* 744
* 745
* 746
* 747
* 748
* 749

* 750
* 751
* 752
* 753
* 754
* 755
* 756
* 757
* 758
* 759

* 760
* 761
* 762
* 763
* 764
* 765
* 766
* 767
* 768
* 769

* 770
* 771
* 772
* 773
* 774
* 775
* 776
* 777
* 778
* 779

* 780
* 781
* 782
* 783
* 784
* 785
* 786
* 787
* 788
* 789

* 790
* 791
* 792
* 793
* 794
* 795
* 796
* 797
* 798
* 799

800s

* 800
* 801
* 802
* 803
* 804
* 805
* 806
* 807
* 808
* 809

* 810
* 811
* 812
* 813
* 814
* 815
* 816
* 817
* 818
* 819

* 820
* 821
* 822
* 823
* 824
* 825
* 826
* 827
* 828
* 829

* 830
* 831
* 832
* 833
* 834
* 835
* 836
* 837
* 838
* 839

* 840
* 841
* 842
* 843
* 844
* 845
* 846
* 847
* 848
* 849

* 850
* 851
* 852
* 853
* 854
* 855
* 856
* 857
* 858
* 859

* 860
* 861
* 862
* 863
* 864
* 865
* 866
* 867
* 868
* 869

* 870
* 871
* 872
* 873
* 874
* 875
* 876
* 877
* 878
* 879

* 880
* 881
* 882
* 883
* 884
* 885
* 886
* 887
* 888
* 889

* 890
* 891
* 892
* 893
* 894
* 895
* 896
* 897
* 898
* 899

900s

* 900
* 901
* 902
* 903
* 904
* 905
* 906
* 907
* 908
* 909

* 910
* 911
* 912
* 913
* 914
* 915
* 916
* 917
* 918
* 919

* 920
* 921
* 922
* 923
* 924
* 925
* 926
* 927
* 928
* 929

* 930
* 931
* 932
* 933
* 934
* 935
* 936
* 937
* 938
* 939

* 940
* 941
* 942
* 943
* 944
* 945
* 946
* 947
* 948
* 949

* 950
* 951
* 952
* 953
* 954
* 955
* 956
* 957
* 958
* 959

* 960
* 961
* 962
* 963
* 964
* 965
* 966
* 967
* 968
* 969

* 970
* 971
* 972
* 973
* 974
* 975
* 976
* 977
* 978
* 979

* 980
* 981
* 982
* 983
* 984
* 985
* 986
* 987
* 988
* 989

* 990
* 991
* 992
* 993
* 994
* 995
* 996
* 997
* 998
* 999

1000s

* 1000
* 1001
* 1023
* 1024
* 1089
* 1093
* 1105
* 1234
* 1289
* 1458
* 1510
* 1728
* 1729
* 1980
* 1987
* 2000
* 2016
* 2520
* 3000
* 3511
* 4000
* 4104
* 5000
* 5040
* 6000
* 6174
* 7000
* 7744
* 7825
* 8000
* 8128
* 8192
* 9000
* 9855
* 9999

* 10,000
* 16,807
* 20,000
* 30,000
* 40,000
* 50,000
* 60,000
* 64,079
* 65,535
* 65,536
* 65,537
* 70,000
* 80,000
* 90,000

* 100,000
* 142,857
* 144,000
* 1,000,000
* 10,000,000
* 43,112,609
* 100,000,000
* 1,000,000,000
* 2,147,483,647
* 4,294,967,295
* 10,000,000,000
* 100,000,000,000

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template