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 |
2 | 371473 | 368918 | 2 |
3 | 246554 | 246970 | 246867 | 3 |
4 | 185925 | 184500 | 185548 | 184418 | 4 |
5 | 148172 | 148141 | 148202 | 147779 | 148097 | 5 |
6 | 123781 | 123188 | 123910 | 122773 | 123782 | 122957 | 6 |
7 | 105594 | 105507 | 105670 | 105850 | 105737 | 106012 | 106021 | 7 |
8 | 93024 | 92383 | 92991 | 92183 | 92901 | 92117 | 92557 | 92235 | 8 |
9 | 81981 | 82298 | 81801 | 81901 | 82244 | 82187 | 82672 | 82428 | 82879 | 9 |
10 | 74284 | 73782 | 74186 | 73403 | 74268 | 73888 | 74359 | 74016 | 74376 | 73829 | 10 |
11 | 67033 | 67337 | 66957 | 67142 | 67377 | 67660 | 67854 | 67179 | 67116 | 67297 | 67439 |
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.