A number whose base 2 representation contains an odd number of ones. more
The first 600 odious numbers :
1,
2,
4,
7,
8,
11,
13,
14,
16,
19,
21,
22,
25,
26,
28,
31,
32,
35,
37,
38,
41,
42,
44,
47,
49,
50,
52,
55,
56,
59,
61,
62,
64,
67,
69,
70,
73,
74,
76,
79,
81,
82,
84,
87,
88,
91,
93,
94,
97,
98,
100,
103,
104,
107,
109,
110,
112,
115,
117,
118,
121,
122,
124,
127,
128,
131,
133,
134,
137,
138,
140,
143,
145,
146,
148,
151,
152,
155,
157,
158,
161,
162,
164,
167,
168,
171,
173,
174,
176,
179,
181,
182,
185,
186,
188,
191,
193,
194,
196,
199,
200,
203,
205,
206,
208,
211,
213,
214,
217,
218,
220,
223,
224,
227,
229,
230,
233,
234,
236,
239,
241,
242,
244,
247,
248,
251,
253,
254,
256,
259,
261,
262,
265,
266,
268,
271,
273,
274,
276,
279,
280,
283,
285,
286,
289,
290,
292,
295,
296,
299,
301,
302,
304,
307,
309,
310,
313,
314,
316,
319,
321,
322,
324,
327,
328,
331,
333,
334,
336,
339,
341,
342,
345,
346,
348,
351,
352,
355,
357,
358,
361,
362,
364,
367,
369,
370,
372,
375,
376,
379,
381,
382,
385,
386,
388,
391,
392,
395,
397,
398,
400,
403,
405,
406,
409,
410,
412,
415,
416,
419,
421,
422,
425,
426,
428,
431,
433,
434,
436,
439,
440,
443,
445,
446,
448,
451,
453,
454,
457,
458,
460,
463,
465,
466,
468,
471,
472,
475,
477,
478,
481,
482,
484,
487,
488,
491,
493,
494,
496,
499,
501,
502,
505,
506,
508,
511,
512,
515,
517,
518,
521,
522,
524,
527,
529,
530,
532,
535,
536,
539,
541,
542,
545,
546,
548,
551,
552,
555,
557,
558,
560,
563,
565,
566,
569,
570,
572,
575,
577,
578,
580,
583,
584,
587,
589,
590,
592,
595,
597,
598,
601,
602,
604,
607,
608,
611,
613,
614,
617,
618,
620,
623,
625,
626,
628,
631,
632,
635,
637,
638,
641,
642,
644,
647,
648,
651,
653,
654,
656,
659,
661,
662,
665,
666,
668,
671,
672,
675,
677,
678,
681,
682,
684,
687,
689,
690,
692,
695,
696,
699,
701,
702,
704,
707,
709,
710,
713,
714,
716,
719,
721,
722,
724,
727,
728,
731,
733,
734,
737,
738,
740,
743,
744,
747,
749,
750,
752,
755,
757,
758,
761,
762,
764,
767,
769,
770,
772,
775,
776,
779,
781,
782,
784,
787,
789,
790,
793,
794,
796,
799,
800,
803,
805,
806,
809,
810,
812,
815,
817,
818,
820,
823,
824,
827,
829,
830,
832,
835,
837,
838,
841,
842,
844,
847,
849,
850,
852,
855,
856,
859,
861,
862,
865,
866,
868,
871,
872,
875,
877,
878,
880,
883,
885,
886,
889,
890,
892,
895,
896,
899,
901,
902,
905,
906,
908,
911,
913,
914,
916,
919,
920,
923,
925,
926,
929,
930,
932,
935,
936,
939,
941,
942,
944,
947,
949,
950,
953,
954,
956,
959,
961,
962,
964,
967,
968,
971,
973,
974,
976,
979,
981,
982,
985,
986,
988,
991,
992,
995,
997,
998,
1001,
1002,
1004,
1007,
1009,
1010,
1012,
1015,
1016,
1019,
1021,
1022,
1024,
1027,
1029,
1030,
1033,
1034,
1036,
1039,
1041,
1042,
1044,
1047,
1048,
1051,
1053,
1054,
1057,
1058,
1060,
1063,
1064,
1067,
1069,
1070,
1072,
1075,
1077,
1078,
1081,
1082,
1084,
1087,
1089,
1090,
1092,
1095,
1096,
1099,
1101,
1102,
1104,
1107,
1109,
1110,
1113,
1114,
1116,
1119,
1120,
1123,
1125,
1126,
1129,
1130,
1132,
1135,
1137,
1138,
1140,
1143,
1144,
1147,
1149,
1150,
1153,
1154,
1156,
1159,
1160,
1163,
1165,
1166,
1168,
1171,
1173,
1174,
1177,
1178,
1180,
1183,
1184,
1187,
1189,
1190,
1193,
1194,
1196,
1199.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 5000001 values, from 1 to 10000001).
n\r | 0 | 1 |
2 | 2500000 | 2500001 | 2 |
3 | 1557344 | 1754997 | 1687660 | 3 |
4 | 1250000 | 1250001 | 1250000 | 1250000 | 4 |
5 | 996565 | 1000286 | 1002630 | 1000000 | 1000520 | 5 |
6 | 767449 | 899218 | 876772 | 789895 | 855779 | 810888 | 6 |
7 | 713579 | 714300 | 714321 | 714800 | 713723 | 714443 | 714835 | 7 |
8 | 625000 | 625001 | 625000 | 625000 | 625000 | 625000 | 625000 | 625000 | 8 |
9 | 519096 | 584996 | 562546 | 519139 | 585008 | 562542 | 519109 | 584993 | 562572 | 9 |
10 | 498090 | 501911 | 501005 | 498995 | 501525 | 498475 | 498375 | 501625 | 501005 | 498995 | 10 |
11 | 454479 | 454603 | 454624 | 454498 | 454578 | 454480 | 454469 | 454612 | 454613 | 454517 | 454528 |
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.