A number n such that 2n + 1 divides 2n + 1. more
The first 600 Curzon numbers :
1,
2,
5,
6,
9,
14,
18,
21,
26,
29,
30,
33,
41,
50,
53,
54,
65,
69,
74,
78,
81,
86,
89,
90,
98,
105,
113,
114,
125,
134,
138,
141,
146,
153,
158,
165,
173,
174,
186,
189,
194,
198,
209,
210,
221,
230,
233,
245,
249,
254,
261,
270,
273,
278,
281,
285,
293,
306,
309,
321,
326,
329,
330,
338,
341,
345,
350,
354,
366,
369,
378,
386,
393,
398,
405,
410,
413,
414,
426,
429,
438,
441,
453,
470,
473,
485,
498,
506,
509,
510,
525,
530,
534,
545,
546,
554,
558,
561,
581,
585,
590,
593,
606,
614,
618,
629,
638,
641,
645,
650,
653,
686,
690,
713,
714,
725,
726,
729,
741,
746,
749,
761,
765,
774,
785,
789,
798,
806,
809,
810,
813,
818,
833,
834,
846,
849,
854,
861,
866,
870,
873,
893,
894,
905,
930,
933,
938,
950,
953,
965,
966,
974,
986,
989,
993,
998,
1001,
1005,
1013,
1014,
1026,
1034,
1041,
1049,
1065,
1070,
1089,
1101,
1106,
1110,
1118,
1121,
1125,
1133,
1134,
1146,
1154,
1166,
1169,
1170,
1173,
1178,
1185,
1190,
1194,
1205,
1218,
1229,
1233,
1238,
1265,
1269,
1274,
1278,
1289,
1310,
1329,
1338,
1341,
1346,
1349,
1353,
1365,
1370,
1374,
1394,
1398,
1401,
1409,
1418,
1421,
1425,
1430,
1454,
1458,
1469,
1478,
1481,
1485,
1505,
1509,
1518,
1530,
1533,
1541,
1554,
1581,
1590,
1593,
1601,
1610,
1614,
1625,
1626,
1629,
1638,
1649,
1650,
1653,
1661,
1665,
1673,
1685,
1686,
1694,
1706,
1730,
1733,
1734,
1745,
1749,
1758,
1766,
1769,
1770,
1773,
1778,
1785,
1790,
1806,
1818,
1821,
1829,
1838,
1845,
1850,
1854,
1866,
1869,
1889,
1898,
1901,
1910,
1925,
1926,
1938,
1953,
1958,
1961,
1965,
1973,
1994,
2001,
2006,
2009,
2010,
2013,
2025,
2045,
2046,
2049,
2066,
2069,
2078,
2105,
2109,
2114,
2121,
2126,
2129,
2130,
2141,
2169,
2174,
2178,
2181,
2186,
2198,
2210,
2225,
2241,
2246,
2253,
2258,
2261,
2273,
2274,
2298,
2301,
2310,
2318,
2321,
2325,
2345,
2361,
2366,
2393,
2394,
2406,
2430,
2438,
2454,
2465,
2466,
2478,
2486,
2493,
2501,
2505,
2510,
2525,
2529,
2538,
2549,
2550,
2553,
2573,
2585,
2589,
2594,
2598,
2613,
2618,
2630,
2654,
2661,
2666,
2673,
2690,
2693,
2706,
2709,
2718,
2721,
2738,
2741,
2750,
2753,
2765,
2778,
2781,
2786,
2790,
2825,
2826,
2829,
2834,
2841,
2846,
2850,
2858,
2870,
2874,
2889,
2906,
2910,
2913,
2921,
2925,
2930,
2933,
2934,
2961,
2969,
2990,
2993,
3005,
3014,
3018,
3021,
3026,
3033,
3045,
3050,
3065,
3066,
3081,
3086,
3098,
3101,
3105,
3110,
3114,
3134,
3138,
3149,
3150,
3158,
3161,
3186,
3189,
3194,
3198,
3210,
3213,
3225,
3234,
3245,
3273,
3281,
3285,
3290,
3309,
3318,
3326,
3329,
3330,
3345,
3350,
3354,
3366,
3381,
3389,
3390,
3401,
3413,
3414,
3434,
3441,
3449,
3453,
3458,
3473,
3474,
3485,
3498,
3506,
3509,
3513,
3521,
3534,
3554,
3593,
3605,
3606,
3609,
3614,
3618,
3621,
3626,
3641,
3653,
3654,
3665,
3666,
3674,
3705,
3725,
3729,
3738,
3749,
3753,
3758,
3761,
3770,
3773,
3774,
3786,
3794,
3801,
3810,
3821,
3834,
3845,
3849,
3858,
3861,
3870,
3878,
3894,
3914,
3926,
3933,
3938,
3941,
3950,
3953,
3966,
3974,
3981,
4005,
4026,
4029,
4034,
4046,
4050,
4058,
4061,
4073,
4085,
4089,
4109,
4110,
4118,
4121,
4134,
4145,
4146,
4158,
4181,
4193,
4194,
4209,
4214,
4221,
4230,
4233,
4250,
4269,
4281,
4286,
4290,
4298,
4313,
4314,
4334,
4338,
4346,
4349,
4353,
4365,
4370,
4373,
4389,
4401,
4409,
4410,
4418,
4430,
4433,
4446,
4461,
4466,
4470,
4481,
4485,
4505,
4506,
4514,
4521,
4529,
4533,
4545,
4554,
4566,
4578,
4586,
4590,
4593,
4601,
4610,
4613,
4638,
4641,
4646,
4661,
4670,
4674,
4685,
4698,
4701,
4706,
4709,
4710,
4718,
4730,
4733,
4745,
4766,
4769.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 10669163 values, from 1 to 199999974).
n\r | 0 | 1 |
2 | 5334824 | 5334339 | 2 |
3 | 5335011 | 1 | 5334151 | 3 |
4 | 0 | 5334339 | 5334824 | 0 | 4 |
5 | 2667365 | 2667736 | 10 | 2666898 | 2667154 | 5 |
6 | 2667506 | 1 | 2667318 | 2667505 | 0 | 2666833 | 6 |
7 | 1778397 | 1778166 | 1778004 | 0 | 1778094 | 1778134 | 1778368 | 7 |
8 | 0 | 2666807 | 2667135 | 0 | 0 | 2667532 | 2667689 | 0 | 8 |
9 | 1778488 | 1 | 1778538 | 1778703 | 0 | 1777348 | 1777820 | 0 | 1778265 | 9 |
10 | 1334035 | 1334078 | 10 | 1333423 | 1333646 | 1333330 | 1333658 | 0 | 1333475 | 1333508 | 10 |
11 | 1067343 | 1066616 | 1066393 | 1066980 | 1067163 | 5 | 1066957 | 1067141 | 1066792 | 1067078 | 1066695 |
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.