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Curzon numbers
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 
253348245334339 2 
3533501115334151 3 
40533433953348240 4 
5266736526677361026668982667154 5 
6266750612667318266750502666833 6 
71778397177816617780040177809417781341778368 7 
802666807266713500266753226676890 8 
9177848811778538177870301777348177782001778265 9 
1013340351334078101333423133364613333301333658013334751333508 10 
1110673431066616106639310669801067163510669571067141106679210670781066695

A pictorial representation of the table above
motab
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.