n is an unusual number iff its largest prime factor is strictly greater than n. The concept of "unusualness" is therefore restricted to the domain of natural numbers. They are in fact a specific example of Finch's rough numbers. A k-rough number has all its prime factors at least k, so an unusual number is √n-rough.

It is obvious that all prime numbers are unusual since their largest prime factor is themselves - certainly greater than their square root! All numbers must have a prime factor less than their square root, if they have a factor at all, but they do not necessarily have a factor greater than the square root.

Schroeppel proved in 1972 that the probability that a randomly chosen number is unusual is ln(2) - not so unusual after all! That is to say, the probability of a number being unusual is independent of the magnitude of the number chosen, as shown in the following table:

n         Unusual numbers < n
10        6
100       67
1000      715
10000     7319
100000    70128

Notice that, as promised, the proportion of unusual numbers up to a certain limit remains remarkably consistent; as n → ∞, the number of unusual numbers → ln(2).

Excluding prime numbers, there are 547 unusual numbers less than 1000:

6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 133, 134, 136, 138, 141, 142, 143, 145, 146, 148, 152, 153, 155, 156, 158, 159, 161, 164, 166, 170, 171, 172, 174, 177, 178, 183, 184, 185, 186, 187, 188, 190, 194, 201, 202, 203, 204, 205, 206, 207, 209, 212, 213, 214, 215, 217, 218, 219, 221, 222, 226, 228, 230, 232, 235, 236, 237, 238, 244, 246, 247, 248, 249, 253, 254, 255, 258, 259, 261, 262, 265, 266, 267, 268, 272, 274, 276, 278, 279, 282, 284, 285, 287, 290, 291, 292, 295, 296, 298, 299, 301, 302, 303, 304, 305, 309, 310, 314, 316, 318, 319, 321, 322, 323, 326, 327, 328, 329, 332, 333, 334, 335, 339, 341, 342, 344, 345, 346, 348, 354, 355, 356, 358, 362, 365, 366, 368, 369, 370, 371, 372, 376, 377, 381, 382, 386, 387, 388, 391, 393, 394, 395, 398, 402, 403, 404, 406, 407, 410, 411, 412, 413, 414, 415, 417, 422, 423, 424, 426, 427, 428, 430, 434, 435, 436, 437, 438, 444, 445, 446, 447, 451, 452, 453, 454, 458, 460, 464, 465, 466, 469, 470, 471, 472, 473, 474, 477, 478, 481, 482, 483, 485, 488, 489, 492, 493, 496, 497, 498, 501, 502, 505, 506, 508, 511, 514, 515, 516, 517, 518, 519, 522, 524, 526, 527, 530, 531, 533, 534, 535, 536, 537, 538, 542, 543, 545, 548, 549, 551, 553, 554, 555, 556, 558, 559, 562, 564, 565, 566, 568, 573, 574, 579, 580, 581, 582, 583, 584, 586, 589, 590, 591, 592, 596, 597, 602, 603, 604, 606, 609, 610, 611, 614, 615, 618, 620, 622, 623, 626, 628, 629, 632, 633, 634, 635, 636, 638, 639, 642, 645, 649, 651, 652, 654, 655, 656, 657, 658, 662, 664, 666, 667, 668, 669, 670, 671, 674, 678, 679, 681, 682, 685, 687, 688, 689, 692, 694, 695, 696, 697, 698, 699, 703, 705, 706, 707, 708, 710, 711, 712, 713, 716, 717, 718, 721, 723, 724, 725, 730, 731, 732, 734, 737, 738, 740, 742, 744, 745, 746, 747, 749, 752, 753, 754, 755, 758, 762, 763, 764, 766, 767, 771, 772, 774, 775, 776, 777, 778, 779, 781, 783, 785, 786, 788, 789, 790, 791, 793, 794, 795, 796, 799, 801, 802, 803, 804, 806, 807, 808, 812, 813, 814, 815, 817, 818, 820, 822, 824, 826, 830, 831, 834, 835, 837, 838, 842, 843, 844, 846, 848, 849, 851, 852, 854, 856, 860, 861, 862, 865, 866, 868, 869, 871, 872, 873, 876, 878, 879, 885, 886, 888, 889, 890, 892, 893, 894, 895, 898, 899, 901, 902, 903, 904, 905, 906, 908, 909, 913, 914, 915, 916, 917, 921, 922, 923, 925, 926, 927, 930, 932, 933, 934, 938, 939, 940, 942, 943, 944, 946, 948, 949, 951, 954, 955, 956, 958, 959, 962, 963, 964, 965, 970, 973, 974, 976, 978, 979, 981, 982, 984, 985, 987, 989, 993, 994, 995, 996, 998, 999.

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