A number n such that n and φ(n) have no common prime factors. more
The first 600 cyclic numbers :
1,
2,
3,
5,
7,
11,
13,
15,
17,
19,
23,
29,
31,
33,
35,
37,
41,
43,
47,
51,
53,
59,
61,
65,
67,
69,
71,
73,
77,
79,
83,
85,
87,
89,
91,
95,
97,
101,
103,
107,
109,
113,
115,
119,
123,
127,
131,
133,
137,
139,
141,
143,
145,
149,
151,
157,
159,
161,
163,
167,
173,
177,
179,
181,
185,
187,
191,
193,
197,
199,
209,
211,
213,
215,
217,
221,
223,
227,
229,
233,
235,
239,
241,
247,
249,
251,
255,
257,
259,
263,
265,
267,
269,
271,
277,
281,
283,
287,
293,
295,
299,
303,
307,
311,
313,
317,
319,
321,
323,
329,
331,
335,
337,
339,
341,
345,
347,
349,
353,
359,
365,
367,
371,
373,
377,
379,
383,
389,
391,
393,
395,
397,
401,
403,
407,
409,
411,
413,
415,
419,
421,
427,
431,
433,
435,
437,
439,
443,
445,
447,
449,
451,
455,
457,
461,
463,
467,
469,
473,
479,
481,
485,
487,
491,
493,
499,
501,
503,
509,
511,
515,
517,
519,
521,
523,
527,
533,
535,
537,
541,
545,
547,
551,
553,
557,
559,
561,
563,
565,
569,
571,
573,
577,
581,
583,
587,
589,
591,
593,
595,
599,
601,
607,
611,
613,
617,
619,
623,
629,
631,
635,
641,
643,
647,
649,
653,
659,
661,
665,
667,
671,
673,
677,
679,
681,
683,
685,
691,
695,
697,
699,
701,
703,
705,
707,
709,
713,
717,
719,
721,
727,
731,
733,
739,
743,
745,
749,
751,
753,
757,
761,
763,
767,
769,
771,
773,
779,
781,
785,
787,
789,
793,
795,
797,
799,
803,
805,
807,
809,
811,
815,
817,
821,
823,
827,
829,
835,
839,
843,
851,
853,
857,
859,
863,
865,
869,
871,
877,
879,
881,
883,
885,
887,
893,
895,
899,
901,
907,
911,
913,
917,
919,
923,
929,
933,
937,
941,
943,
947,
949,
951,
953,
957,
959,
965,
967,
971,
973,
977,
983,
985,
989,
991,
995,
997,
1001,
1003,
1007,
1009,
1013,
1019,
1021,
1031,
1033,
1037,
1039,
1041,
1043,
1049,
1051,
1057,
1059,
1061,
1063,
1067,
1069,
1073,
1077,
1079,
1087,
1091,
1093,
1097,
1099,
1103,
1105,
1109,
1111,
1115,
1117,
1121,
1123,
1129,
1133,
1135,
1139,
1141,
1145,
1147,
1149,
1151,
1153,
1157,
1159,
1163,
1165,
1167,
1169,
1171,
1173,
1177,
1181,
1187,
1189,
1193,
1195,
1199,
1201,
1203,
1207,
1211,
1213,
1217,
1219,
1223,
1229,
1231,
1235,
1237,
1241,
1243,
1245,
1247,
1249,
1253,
1257,
1259,
1261,
1267,
1271,
1273,
1277,
1279,
1283,
1285,
1289,
1291,
1293,
1295,
1297,
1301,
1303,
1307,
1309,
1313,
1315,
1319,
1321,
1327,
1329,
1333,
1335,
1337,
1339,
1343,
1345,
1347,
1349,
1351,
1353,
1357,
1361,
1363,
1367,
1373,
1381,
1383,
1385,
1387,
1391,
1393,
1397,
1399,
1401,
1403,
1409,
1411,
1415,
1417,
1423,
1427,
1429,
1433,
1437,
1439,
1441,
1447,
1451,
1453,
1457,
1459,
1463,
1465,
1469,
1471,
1473,
1479,
1481,
1483,
1487,
1489,
1493,
1495,
1499,
1501,
1507,
1509,
1511,
1513,
1517,
1523,
1527,
1529,
1531,
1535,
1537,
1541,
1543,
1547,
1549,
1551,
1553,
1559,
1561,
1563,
1565,
1567,
1571,
1577,
1579,
1583,
1585,
1589,
1591,
1597,
1601,
1603,
1605,
1607,
1609,
1613,
1615,
1619,
1621,
1627,
1631,
1633,
1637,
1639,
1643,
1645,
1649,
1651,
1657,
1661,
1663,
1667,
1669,
1671,
1679,
1685,
1687,
1689,
1691,
1693,
1695,
1697,
1699,
1707,
1709,
1717,
1721,
1723,
1727,
1729,
1733,
1735,
1739,
1741,
1745,
1747,
1749,
1753,
1757,
1759,
1761,
1763,
1765,
1769,
1777,
1779,
1781,
1783,
1787,
1789,
1793,
1795,
1797,
1799,
1801,
1807,
1811,
1817,
1819,
1823,
1829,
1831,
1835,
1837,
1841,
1843,
1847,
1851,
1853,
1855,
1861,
1865,
1867,
1871,
1873.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 2892408 values, from 1 to 9999997).
n\r | 0 | 1 |
2 | 1 | 2892407 | 2 |
3 | 260049 | 1316150 | 1316209 | 3 |
4 | 0 | 1446117 | 1 | 1446290 | 4 |
5 | 312765 | 644807 | 645017 | 644829 | 644990 | 5 |
6 | 0 | 1316150 | 1 | 260049 | 0 | 1316208 | 6 |
7 | 255791 | 439409 | 439500 | 439418 | 439417 | 439493 | 439380 | 7 |
8 | 0 | 723169 | 1 | 723084 | 0 | 722948 | 0 | 723206 | 8 |
9 | 0 | 438729 | 438786 | 114672 | 438756 | 438793 | 145377 | 438665 | 438630 | 9 |
10 | 0 | 644807 | 1 | 644829 | 0 | 312765 | 0 | 645016 | 0 | 644990 | 10 |
11 | 184214 | 270739 | 270793 | 270918 | 270674 | 270849 | 270771 | 270926 | 270902 | 270793 | 270829 |
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.