*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.