A number whose base 2 representation contains a prime number of ones. more
The first 600 pernicious numbers :
3,
5,
6,
7,
9,
10,
11,
12,
13,
14,
17,
18,
19,
20,
21,
22,
24,
25,
26,
28,
31,
33,
34,
35,
36,
37,
38,
40,
41,
42,
44,
47,
48,
49,
50,
52,
55,
56,
59,
61,
62,
65,
66,
67,
68,
69,
70,
72,
73,
74,
76,
79,
80,
81,
82,
84,
87,
88,
91,
93,
94,
96,
97,
98,
100,
103,
104,
107,
109,
110,
112,
115,
117,
118,
121,
122,
124,
127,
129,
130,
131,
132,
133,
134,
136,
137,
138,
140,
143,
144,
145,
146,
148,
151,
152,
155,
157,
158,
160,
161,
162,
164,
167,
168,
171,
173,
174,
176,
179,
181,
182,
185,
186,
188,
191,
192,
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,
257,
258,
259,
260,
261,
262,
264,
265,
266,
268,
271,
272,
273,
274,
276,
279,
280,
283,
285,
286,
288,
289,
290,
292,
295,
296,
299,
301,
302,
304,
307,
309,
310,
313,
314,
316,
319,
320,
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,
384,
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,
513,
514,
515,
516,
517,
518,
520,
521,
522,
524,
527,
528,
529,
530,
532,
535,
536,
539,
541,
542,
544,
545,
546,
548,
551,
552,
555,
557,
558,
560,
563,
565,
566,
569,
570,
572,
575,
576,
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,
640,
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,
768,
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,
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,
961,
962,
964,
967,
968,
971,
973,
974,
976,
979,
981,
982,
985,
986,
988,
992,
995,
997,
998,
1001,
1002,
1004,
1009,
1010,
1012,
1016,
1025,
1026,
1027,
1028,
1029,
1030,
1032,
1033,
1034,
1036,
1039,
1040,
1041,
1042,
1044,
1047,
1048,
1051,
1053,
1054,
1056,
1057,
1058,
1060,
1063,
1064,
1067,
1069,
1070,
1072,
1075,
1077,
1078,
1081,
1082,
1084,
1087,
1088,
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.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 3443660 values, from 3 to 9999998).
n\r | 0 | 1 |
2 | 1700886 | 1742774 | 2 |
3 | 1074200 | 1207543 | 1161917 | 3 |
4 | 831270 | 869616 | 869616 | 873158 | 4 |
5 | 686422 | 688926 | 690494 | 688736 | 689082 | 5 |
6 | 522668 | 625433 | 596108 | 551532 | 582110 | 565809 | 6 |
7 | 468531 | 497758 | 499010 | 501043 | 490709 | 490040 | 496569 | 7 |
8 | 403621 | 427649 | 427649 | 441967 | 427649 | 441967 | 441967 | 431191 | 8 |
9 | 358062 | 402504 | 387300 | 358077 | 402528 | 387301 | 358061 | 402511 | 387316 | 9 |
10 | 338888 | 349838 | 340854 | 347882 | 341202 | 347534 | 339088 | 349640 | 340854 | 347880 | 10 |
11 | 313017 | 313099 | 313112 | 313027 | 313083 | 313015 | 313007 | 313107 | 313106 | 313039 | 313048 |
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.