A number n for which the number of digits in its prime factorization is larger than its number of digits. more
The first 600 wasteful numbers :
4,
6,
8,
9,
12,
18,
20,
22,
24,
26,
28,
30,
33,
34,
36,
38,
39,
40,
42,
44,
45,
46,
48,
50,
51,
52,
54,
55,
56,
57,
58,
60,
62,
63,
65,
66,
68,
69,
70,
72,
74,
75,
76,
77,
78,
80,
82,
84,
85,
86,
87,
88,
90,
91,
92,
93,
94,
95,
96,
98,
99,
100,
102,
104,
108,
110,
114,
116,
117,
120,
124,
126,
130,
132,
136,
138,
140,
143,
144,
148,
150,
152,
153,
154,
156,
164,
165,
168,
170,
171,
172,
174,
176,
180,
182,
184,
186,
187,
188,
190,
195,
196,
198,
200,
202,
204,
206,
207,
208,
209,
210,
212,
214,
216,
218,
220,
221,
222,
225,
226,
228,
230,
231,
232,
234,
236,
238,
240,
242,
244,
246,
247,
248,
252,
253,
254,
255,
258,
260,
261,
262,
264,
266,
268,
270,
272,
273,
274,
275,
276,
278,
279,
280,
282,
284,
285,
286,
288,
290,
292,
294,
296,
297,
298,
299,
300,
302,
303,
304,
306,
308,
309,
310,
312,
314,
315,
316,
318,
319,
321,
322,
323,
324,
325,
326,
327,
328,
330,
332,
333,
334,
336,
338,
339,
340,
341,
342,
344,
345,
346,
348,
350,
351,
352,
354,
356,
357,
358,
360,
362,
363,
364,
366,
368,
369,
370,
372,
374,
376,
377,
378,
380,
381,
382,
385,
386,
387,
388,
390,
391,
392,
393,
394,
396,
398,
399,
400,
402,
403,
404,
406,
407,
408,
410,
411,
412,
414,
416,
417,
418,
420,
422,
423,
424,
425,
426,
428,
429,
430,
432,
434,
435,
436,
437,
438,
440,
441,
442,
444,
446,
447,
450,
451,
452,
453,
454,
455,
456,
458,
459,
460,
462,
464,
465,
466,
468,
470,
471,
472,
473,
474,
475,
476,
477,
478,
480,
481,
482,
483,
484,
488,
489,
490,
492,
493,
494,
495,
496,
498,
500,
501,
502,
504,
505,
506,
507,
508,
510,
513,
514,
515,
516,
517,
518,
519,
520,
522,
524,
525,
526,
527,
528,
530,
531,
532,
533,
534,
535,
536,
537,
538,
539,
540,
542,
543,
544,
545,
546,
548,
549,
550,
551,
552,
554,
555,
556,
558,
559,
560,
561,
562,
564,
565,
566,
568,
570,
572,
573,
574,
575,
576,
578,
579,
580,
582,
583,
584,
585,
586,
588,
589,
590,
591,
592,
594,
595,
596,
597,
598,
600,
602,
603,
604,
605,
606,
608,
609,
610,
611,
612,
614,
615,
616,
618,
620,
621,
622,
624,
626,
627,
628,
629,
630,
632,
633,
634,
635,
636,
637,
638,
639,
642,
644,
645,
646,
648,
649,
650,
651,
652,
654,
655,
656,
657,
658,
660,
662,
663,
664,
665,
666,
667,
668,
669,
670,
671,
672,
674,
675,
676,
678,
680,
681,
682,
684,
685,
687,
688,
689,
690,
692,
693,
694,
695,
696,
697,
698,
699,
700,
702,
703,
704,
705,
706,
707,
708,
710,
711,
712,
713,
714,
715,
716,
717,
718,
720,
721,
722,
723,
724,
725,
726,
728,
730,
731,
732,
734,
735,
736,
737,
738,
740,
741,
742,
744,
745,
746,
747,
748,
749,
750,
752,
753,
754,
755,
756,
758,
759,
760,
762,
763,
764,
765,
766,
767,
770,
771,
772,
774,
775,
776,
777,
778,
779,
780,
781,
782,
783,
784,
785,
786,
788,
789,
790,
791,
792,
793,
794,
795,
796,
798,
799,
800,
801,
802,
803,
804,
805,
806,
807,
808,
810,
812,
813,
814,
815,
816,
817,
818,
819,
820,
822,
824,
825,
826,
828,
830,
831,
832,
833,
834,
835,
836,
837,
838,
840,
842,
843,
844,
845,
846,
847,
848,
849,
850,
851,
852,
854,
855,
856.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 8498885 values, from 4 to 9999999).
n\r | 0 | 1 |
2 | 4854537 | 3644348 | 2 |
3 | 3185666 | 2656583 | 2656636 | 3 |
4 | 2421887 | 1822134 | 2432650 | 1822214 | 4 |
5 | 1843797 | 1663756 | 1663777 | 1663712 | 1663843 | 5 |
6 | 1648311 | 1053402 | 1603045 | 1537355 | 1603181 | 1053591 | 6 |
7 | 1264668 | 1205736 | 1205639 | 1205684 | 1205650 | 1205712 | 1205796 | 7 |
8 | 1174133 | 910989 | 1216275 | 911014 | 1247754 | 911145 | 1216375 | 911200 | 8 |
9 | 1059131 | 885492 | 885492 | 1063246 | 885499 | 885607 | 1063289 | 885592 | 885537 | 9 |
10 | 984146 | 696121 | 967569 | 696155 | 967630 | 859651 | 967635 | 696208 | 967557 | 696213 | 10 |
11 | 901825 | 759670 | 759669 | 759664 | 759782 | 759696 | 759730 | 759627 | 759670 | 759755 | 759797 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.