Composite numbers which have no factors in common with the sum of their divisors. more
The first 600 Duffinian numbers :
4,
8,
9,
16,
21,
25,
27,
32,
35,
36,
39,
49,
50,
55,
57,
63,
64,
65,
75,
77,
81,
85,
93,
98,
100,
111,
115,
119,
121,
125,
128,
129,
133,
143,
144,
155,
161,
169,
171,
175,
183,
185,
187,
189,
201,
203,
205,
209,
215,
217,
219,
221,
225,
235,
237,
242,
243,
245,
247,
253,
256,
259,
265,
275,
279,
289,
291,
299,
301,
305,
309,
319,
323,
324,
325,
327,
329,
333,
335,
338,
341,
343,
351,
355,
361,
363,
365,
371,
377,
381,
385,
387,
391,
392,
399,
400,
403,
407,
413,
415,
417,
425,
427,
437,
451,
453,
469,
471,
481,
484,
485,
489,
493,
497,
505,
511,
512,
513,
515,
517,
525,
527,
529,
533,
535,
539,
543,
549,
551,
553,
559,
565,
567,
575,
576,
578,
579,
583,
589,
595,
597,
603,
605,
611,
623,
625,
629,
633,
635,
649,
651,
655,
657,
667,
669,
671,
676,
685,
687,
689,
697,
707,
711,
713,
715,
721,
722,
723,
729,
731,
737,
741,
749,
755,
763,
767,
777,
779,
781,
784,
785,
791,
793,
799,
800,
803,
805,
813,
815,
817,
831,
833,
835,
837,
841,
845,
849,
851,
865,
867,
869,
871,
873,
875,
889,
893,
899,
900,
901,
903,
905,
913,
917,
921,
923,
925,
927,
935,
939,
943,
949,
955,
959,
961,
965,
968,
975,
979,
981,
985,
989,
993,
999,
1003,
1007,
1011,
1024,
1025,
1027,
1029,
1037,
1043,
1047,
1053,
1055,
1057,
1058,
1067,
1073,
1075,
1079,
1081,
1085,
1089,
1099,
1101,
1105,
1111,
1115,
1119,
1121,
1127,
1133,
1135,
1137,
1141,
1143,
1147,
1156,
1157,
1159,
1161,
1165,
1175,
1177,
1183,
1189,
1191,
1197,
1205,
1207,
1209,
1211,
1219,
1225,
1227,
1241,
1243,
1247,
1250,
1251,
1253,
1255,
1261,
1263,
1265,
1271,
1273,
1281,
1285,
1295,
1296,
1299,
1309,
1313,
1315,
1317,
1325,
1331,
1333,
1337,
1343,
1349,
1351,
1352,
1355,
1357,
1359,
1363,
1369,
1371,
1375,
1379,
1385,
1387,
1389,
1391,
1393,
1397,
1403,
1405,
1407,
1411,
1413,
1415,
1417,
1421,
1443,
1444,
1445,
1457,
1461,
1463,
1465,
1467,
1469,
1477,
1495,
1497,
1501,
1505,
1507,
1513,
1517,
1519,
1525,
1529,
1533,
1535,
1537,
1539,
1541,
1555,
1565,
1569,
1573,
1575,
1577,
1585,
1587,
1589,
1591,
1600,
1603,
1623,
1629,
1631,
1633,
1639,
1641,
1643,
1645,
1647,
1649,
1651,
1655,
1659,
1661,
1673,
1675,
1677,
1679,
1681,
1682,
1685,
1687,
1691,
1703,
1705,
1711,
1713,
1727,
1731,
1735,
1737,
1739,
1751,
1763,
1765,
1767,
1769,
1771,
1775,
1781,
1791,
1793,
1799,
1803,
1805,
1807,
1809,
1813,
1817,
1819,
1821,
1825,
1829,
1835,
1837,
1839,
1841,
1843,
1849,
1853,
1855,
1857,
1859,
1865,
1875,
1883,
1893,
1897,
1899,
1903,
1909,
1915,
1919,
1921,
1922,
1925,
1927,
1929,
1936,
1937,
1939,
1943,
1953,
1955,
1957,
1961,
1963,
1967,
1969,
1971,
1981,
1983,
1985,
1991,
2005,
2007,
2015,
2019,
2021,
2023,
2025,
2033,
2035,
2041,
2047,
2048,
2057,
2059,
2061,
2071,
2073,
2075,
2077,
2101,
2105,
2107,
2116,
2117,
2119,
2123,
2125,
2127,
2133,
2135,
2155,
2159,
2163,
2165,
2169,
2171,
2173,
2177,
2181,
2183,
2187,
2189,
2191,
2197,
2199,
2201,
2209,
2215,
2217,
2219,
2227,
2231,
2233,
2249,
2253,
2255,
2257,
2261,
2263,
2271,
2279,
2285,
2289,
2291,
2303,
2304,
2305,
2307,
2312,
2315,
2317,
2321,
2323,
2327,
2329,
2331,
2335,
2345,
2359,
2361,
2363,
2369,
2379,
2387,
2401,
2405,
2407,
2413,
2419,
2425,
2429,
2431,
2433,
2435,
2439,
2449,
2450,
2451,
2453,
2455,
2461,
2469,
2471,
2479,
2483,
2485,
2487,
2489,
2491,
2493,
2497,
2500,
2501,
2507,
2509,
2511,
2513,
2515,
2519,
2523,
2525,
2527,
2533,
2537,
2547,
2555,
2559,
2561,
2563,
2567,
2569,
2573.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 2543279 values, from 4 to 9999997).
n\r | 0 | 1 |
2 | 2450 | 2540829 | 2 |
3 | 372851 | 1085074 | 1085354 | 3 |
4 | 1762 | 1270710 | 688 | 1270119 | 4 |
5 | 395961 | 536836 | 536756 | 536736 | 536990 | 5 |
6 | 266 | 1084156 | 1266 | 372585 | 918 | 1084088 | 6 |
7 | 340801 | 367634 | 367462 | 366632 | 367368 | 366615 | 366767 | 7 |
8 | 1199 | 635355 | 688 | 635065 | 563 | 635355 | 0 | 635054 | 8 |
9 | 124950 | 361693 | 361766 | 247901 | 361695 | 361809 | 0 | 361686 | 361779 | 9 |
10 | 418 | 536362 | 545 | 536190 | 467 | 395543 | 474 | 536211 | 546 | 536523 | 10 |
11 | 226552 | 231785 | 231580 | 231610 | 231637 | 231834 | 231588 | 231649 | 231672 | 231693 | 231679 |
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.