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Ulam numbers
Members of the sequence defined by U(1)=1, U(2)=2 and U(k) equal to the smallest number which can be written in exactly one way a sum of two distinct previous terms. more

The first 600 Ulam numbers :
1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126, 131, 138, 145, 148, 155, 175, 177, 180, 182, 189, 197, 206, 209, 219, 221, 236, 238, 241, 243, 253, 258, 260, 273, 282, 309, 316, 319, 324, 339, 341, 356, 358, 363, 370, 382, 390, 400, 402, 409, 412, 414, 429, 431, 434, 441, 451, 456, 483, 485, 497, 502, 522, 524, 544, 546, 566, 568, 585, 602, 605, 607, 612, 624, 627, 646, 668, 673, 685, 688, 690, 695, 720, 722, 732, 734, 739, 751, 781, 783, 798, 800, 820, 847, 849, 861, 864, 866, 891, 893, 905, 927, 949, 983, 986, 991, 1018, 1020, 1023, 1025, 1030, 1032, 1035, 1037, 1052, 1079, 1081, 1101, 1103, 1125, 1155, 1157, 1164, 1167, 1169, 1186, 1191, 1208, 1230, 1252, 1257, 1296, 1308, 1311, 1313, 1335, 1338, 1340, 1355, 1360, 1377, 1387, 1389, 1404, 1406, 1428, 1431, 1433, 1462, 1465, 1470, 1472, 1489, 1492, 1509, 1514, 1516, 1531, 1536, 1538, 1550, 1553, 1594, 1602, 1604, 1616, 1641, 1643, 1646, 1648, 1660, 1682, 1707, 1709, 1721, 1724, 1748, 1765, 1770, 1790, 1792, 1812, 1814, 1834, 1836, 1853, 1856, 1858, 1900, 1902, 1919, 1941, 1944, 1946, 1966, 1968, 1985, 2010, 2012, 2032, 2034, 2054, 2056, 2090, 2093, 2095, 2112, 2115, 2117, 2134, 2156, 2178, 2247, 2249, 2252, 2254, 2288, 2327, 2330, 2332, 2354, 2371, 2393, 2418, 2420, 2445, 2447, 2462, 2464, 2481, 2484, 2486, 2511, 2513, 2525, 2550, 2552, 2572, 2574, 2581, 2584, 2589, 2613, 2616, 2618, 2628, 2630, 2633, 2635, 2650, 2660, 2662, 2674, 2696, 2721, 2723, 2748, 2750, 2762, 2787, 2789, 2809, 2811, 2814, 2816, 2831, 2833, 2897, 2899, 2916, 2919, 2921, 2985, 2987, 3029, 3031, 3038, 3041, 3043, 3065, 3068, 3070, 3085, 3090, 3092, 3107, 3109, 3131, 3153, 3205, 3207, 3214, 3217, 3219, 3236, 3239, 3261, 3263, 3288, 3290, 3305, 3368, 3371, 3373, 3390, 3393, 3395, 3415, 3417, 3451, 3454, 3456, 3473, 3476, 3481, 3483, 3495, 3525, 3527, 3544, 3547, 3549, 3591, 3593, 3605, 3608, 3610, 3622, 3625, 3630, 3632, 3649, 3669, 3671, 3691, 3696, 3698, 3723, 3725, 3740, 3742, 3759, 3762, 3764, 3806, 3808, 3825, 3872, 3874, 3886, 3916, 3918, 3930, 3952, 3960, 3962, 3974, 3991, 3994, 4018, 4038, 4040, 4057, 4101, 4118, 4121, 4148, 4150, 4153, 4155, 4165, 4167, 4187, 4211, 4233, 4258, 4260, 4294, 4297, 4324, 4326, 4341, 4343, 4363, 4365, 4368, 4370, 4390, 4392, 4404, 4407, 4409, 4451, 4453, 4470, 4517, 4519, 4531, 4534, 4536, 4578, 4580, 4600, 4602, 4619, 4622, 4624, 4641, 4644, 4646, 4666, 4668, 4707, 4729, 4732, 4734, 4754, 4756, 4798, 4800, 4878, 4881, 4883, 4900, 4903, 4905, 4925, 4927, 4969, 4971, 4996, 4998, 5018, 5020, 5032, 5035, 5037, 5049, 5052, 5057, 5059, 5079, 5081, 5096, 5118, 5159, 5162, 5164, 5181, 5184, 5186, 5206, 5208, 5250, 5252, 5269, 5272, 5274, 5291, 5294, 5296, 5316, 5318, 5335, 5357, 5382, 5384, 5484, 5487, 5489, 5514, 5516, 5531, 5533, 5550, 5597, 5599, 5616, 5663, 5665, 5685, 5687, 5765, 5768, 5770, 5795, 5797, 5812, 5814, 5826, 5829, 5853, 5856, 5858, 5873, 5875, 5878, 5880, 5895, 5900, 5902, 5944, 5946, 6024, 6027, 6029, 6046, 6049, 6051, 6068, 6107, 6110, 6112, 6134, 6154, 6156, 6176, 6178, 6239, 6242, 6244, 6283, 6308, 6310, 6322, 6325, 6327, 6352, 6354, 6366, 6391, 6393, 6410, 6418, 6420, 6435, 6437, 6457, 6459, 6479, 6481, 6520, 6523, 6525, 6550, 6552, 6567, 6569, 6586, 6633, 6635, 6652, 6721, 6723, 6735, 6738, 6740, 6782, 6784, 6804, 6806, 6831, 6833, 6862, 6865, 6870, 6872, 6889, 6892, 6894, 6909, 6911, 6914, 6916, 6931, 6936, 6938, 6980, 6982, 7038, 7041.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 740391 values, from 1 to 10000241).

n\r 0  1 
2371473368918 2 
3246554246970246867 3 
4185925184500185548184418 4 
5148172148141148202147779148097 5 
6123781123188123910122773123782122957 6 
7105594105507105670105850105737106012106021 7 
89302492383929919218392901921179255792235 8 
9819818229881801819018224482187826728242882879 9 
1074284737827418673403742687388874359740167437673829 10 
116703367337669576714267377676606785467179671166729767439

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.