A number whose base 2 representation contains an even number of ones. more
The first 600 evil numbers :
3,
5,
6,
9,
10,
12,
15,
17,
18,
20,
23,
24,
27,
29,
30,
33,
34,
36,
39,
40,
43,
45,
46,
48,
51,
53,
54,
57,
58,
60,
63,
65,
66,
68,
71,
72,
75,
77,
78,
80,
83,
85,
86,
89,
90,
92,
95,
96,
99,
101,
102,
105,
106,
108,
111,
113,
114,
116,
119,
120,
123,
125,
126,
129,
130,
132,
135,
136,
139,
141,
142,
144,
147,
149,
150,
153,
154,
156,
159,
160,
163,
165,
166,
169,
170,
172,
175,
177,
178,
180,
183,
184,
187,
189,
190,
192,
195,
197,
198,
201,
202,
204,
207,
209,
210,
212,
215,
216,
219,
221,
222,
225,
226,
228,
231,
232,
235,
237,
238,
240,
243,
245,
246,
249,
250,
252,
255,
257,
258,
260,
263,
264,
267,
269,
270,
272,
275,
277,
278,
281,
282,
284,
287,
288,
291,
293,
294,
297,
298,
300,
303,
305,
306,
308,
311,
312,
315,
317,
318,
320,
323,
325,
326,
329,
330,
332,
335,
337,
338,
340,
343,
344,
347,
349,
350,
353,
354,
356,
359,
360,
363,
365,
366,
368,
371,
373,
374,
377,
378,
380,
383,
384,
387,
389,
390,
393,
394,
396,
399,
401,
402,
404,
407,
408,
411,
413,
414,
417,
418,
420,
423,
424,
427,
429,
430,
432,
435,
437,
438,
441,
442,
444,
447,
449,
450,
452,
455,
456,
459,
461,
462,
464,
467,
469,
470,
473,
474,
476,
479,
480,
483,
485,
486,
489,
490,
492,
495,
497,
498,
500,
503,
504,
507,
509,
510,
513,
514,
516,
519,
520,
523,
525,
526,
528,
531,
533,
534,
537,
538,
540,
543,
544,
547,
549,
550,
553,
554,
556,
559,
561,
562,
564,
567,
568,
571,
573,
574,
576,
579,
581,
582,
585,
586,
588,
591,
593,
594,
596,
599,
600,
603,
605,
606,
609,
610,
612,
615,
616,
619,
621,
622,
624,
627,
629,
630,
633,
634,
636,
639,
640,
643,
645,
646,
649,
650,
652,
655,
657,
658,
660,
663,
664,
667,
669,
670,
673,
674,
676,
679,
680,
683,
685,
686,
688,
691,
693,
694,
697,
698,
700,
703,
705,
706,
708,
711,
712,
715,
717,
718,
720,
723,
725,
726,
729,
730,
732,
735,
736,
739,
741,
742,
745,
746,
748,
751,
753,
754,
756,
759,
760,
763,
765,
766,
768,
771,
773,
774,
777,
778,
780,
783,
785,
786,
788,
791,
792,
795,
797,
798,
801,
802,
804,
807,
808,
811,
813,
814,
816,
819,
821,
822,
825,
826,
828,
831,
833,
834,
836,
839,
840,
843,
845,
846,
848,
851,
853,
854,
857,
858,
860,
863,
864,
867,
869,
870,
873,
874,
876,
879,
881,
882,
884,
887,
888,
891,
893,
894,
897,
898,
900,
903,
904,
907,
909,
910,
912,
915,
917,
918,
921,
922,
924,
927,
928,
931,
933,
934,
937,
938,
940,
943,
945,
946,
948,
951,
952,
955,
957,
958,
960,
963,
965,
966,
969,
970,
972,
975,
977,
978,
980,
983,
984,
987,
989,
990,
993,
994,
996,
999,
1000,
1003,
1005,
1006,
1008,
1011,
1013,
1014,
1017,
1018,
1020,
1023,
1025,
1026,
1028,
1031,
1032,
1035,
1037,
1038,
1040,
1043,
1045,
1046,
1049,
1050,
1052,
1055,
1056,
1059,
1061,
1062,
1065,
1066,
1068,
1071,
1073,
1074,
1076,
1079,
1080,
1083,
1085,
1086,
1088,
1091,
1093,
1094,
1097,
1098,
1100,
1103,
1105,
1106,
1108,
1111,
1112,
1115,
1117,
1118,
1121,
1122,
1124,
1127,
1128,
1131,
1133,
1134,
1136,
1139,
1141,
1142,
1145,
1146,
1148,
1151,
1152,
1155,
1157,
1158,
1161,
1162,
1164,
1167,
1169,
1170,
1172,
1175,
1176,
1179,
1181,
1182,
1185,
1186,
1188,
1191,
1192,
1195,
1197,
1198,
1200.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 5000000 values, from 3 to 10000000).
n\r | 0 | 1 |
2 | 2500000 | 2500000 | 2 |
3 | 1775989 | 1578337 | 1645674 | 3 |
4 | 1250000 | 1250000 | 1250000 | 1250000 | 4 |
5 | 1003435 | 999715 | 997370 | 1000000 | 999480 | 5 |
6 | 899217 | 767449 | 789895 | 876772 | 810888 | 855779 | 6 |
7 | 714992 | 714272 | 714251 | 713772 | 714849 | 714128 | 713736 | 7 |
8 | 625000 | 625000 | 625000 | 625000 | 625000 | 625000 | 625000 | 625000 | 8 |
9 | 592015 | 526116 | 548566 | 591972 | 526103 | 548569 | 592002 | 526118 | 548539 | 9 |
10 | 501910 | 498090 | 498995 | 501005 | 498475 | 501525 | 501625 | 498375 | 498995 | 501005 | 10 |
11 | 454612 | 454488 | 454467 | 454593 | 454513 | 454611 | 454622 | 454479 | 454478 | 454574 | 454563 |
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.