A number which can be nontrivially obtained combining all its digits with the basic operations and concatenation. more
The first 600 Friedman numbers :
25,
121,
125,
126,
127,
128,
153,
216,
289,
343,
347,
625,
688,
736,
1022,
1024,
1206,
1255,
1260,
1285,
1296,
1395,
1435,
1503,
1530,
1792,
1827,
2048,
2187,
2349,
2500,
2501,
2502,
2503,
2504,
2505,
2506,
2507,
2508,
2509,
2592,
2737,
2916,
3125,
3159,
3281,
3375,
3378,
3685,
3784,
3864,
3972,
4088,
4096,
4106,
4167,
4536,
4624,
4628,
5120,
5776,
5832,
6144,
6145,
6455,
6880,
7928,
8092,
8192,
9025,
9216,
9261,
10192,
10201,
10251,
10255,
10368,
10426,
10521,
10525,
10575,
10824,
10935,
11025,
11163,
11259,
11264,
11439,
11663,
11664,
11665,
11844,
11848,
11943,
12006,
12060,
12091,
12100,
12101,
12102,
12103,
12104,
12105,
12106,
12107,
12108,
12109,
12167,
12288,
12321,
12337,
12384,
12493,
12505,
12544,
12546,
12550,
12595,
12600,
12762,
12768,
12769,
12798,
12802,
12843,
12850,
12955,
12960,
12964,
12996,
13125,
13176,
13225,
13243,
13286,
13496,
13545,
13689,
13725,
13764,
13813,
13822,
13823,
13824,
13825,
13826,
13832,
13842,
13950,
14035,
14129,
14168,
14175,
14256,
14350,
14352,
14641,
14645,
14647,
15003,
15030,
15125,
15246,
15300,
15345,
15378,
15379,
15435,
15495,
15552,
15562,
15567,
15568,
15585,
15586,
15612,
15613,
15615,
15617,
15618,
15620,
15621,
15622,
15623,
15624,
15625,
15626,
15627,
15628,
15629,
15631,
15632,
15633,
15634,
15635,
15641,
15642,
15645,
15655,
15656,
15661,
15662,
15667,
15679,
15688,
15689,
15697,
15698,
15795,
15975,
16225,
16245,
16272,
16295,
16347,
16348,
16368,
16372,
16374,
16375,
16377,
16378,
16381,
16382,
16384,
16385,
16387,
16447,
16448,
16479,
16743,
16758,
16759,
16765,
16783,
16794,
16797,
16798,
16807,
16815,
16875,
16879,
17253,
17325,
17328,
17346,
17368,
17384,
17428,
17437,
17482,
17488,
17536,
17689,
17856,
17892,
17920,
17925,
18225,
18265,
18270,
18432,
18435,
18522,
18594,
18723,
18744,
19026,
19215,
19321,
19392,
19453,
19592,
19629,
19642,
19653,
19682,
19683,
19684,
19692,
19693,
19732,
19734,
19736,
19737,
19738,
19739,
19773,
19845,
20485,
20736,
21175,
21375,
21495,
21504,
21586,
21606,
21753,
21843,
21844,
21845,
21848,
21870,
21875,
21943,
21952,
21953,
22264,
22528,
22757,
23326,
23328,
23392,
23456,
23490,
23546,
23548,
23552,
23796,
24336,
24339,
24367,
24375,
24385,
24389,
24390,
24393,
24546,
24564,
24566,
24576,
24584,
24586,
24768,
24964,
24972,
25105,
25137,
25314,
25375,
25474,
25510,
25725,
25872,
25895,
25921,
26238,
26244,
26348,
26364,
26496,
26624,
26754,
26896,
26973,
27436,
27634,
27639,
27648,
27653,
27654,
27783,
27889,
28217,
28224,
28226,
28322,
28476,
28547,
28554,
28556,
28559,
28561,
28564,
28671,
28672,
28674,
28678,
28728,
28749,
28764,
28784,
28900,
29160,
29184,
29282,
29517,
29519,
29523,
29524,
29525,
29526,
29527,
29529,
29531,
29549,
29584,
29632,
29768,
29795,
29929,
30625,
31250,
31252,
31256,
31346,
31347,
31509,
31590,
31682,
32685,
32697,
32744,
32747,
32751,
32759,
32761,
32762,
32764,
32765,
32768,
32771,
32772,
32775,
32778,
32781,
32782,
32783,
32785,
32786,
32795,
32805,
32815,
32825,
32832,
32835,
32836,
32845,
32849,
32851,
32853,
32854,
32855,
32859,
32865,
32875,
32885,
32895,
33495,
33579,
33655,
33696,
34425,
34968,
34986,
34991,
34992,
34993,
34996,
35152,
35684,
35721,
35726,
35782,
35928,
35932,
35933,
35937,
35942,
36457,
36549,
36850,
36855,
36864,
36918,
37179,
37187,
37249,
37449,
37668,
37814,
37840,
37845,
37875,
38416,
38424,
38427,
38637,
38640,
38856,
38912,
39216,
39283,
39288,
39294,
39295,
39304,
39313,
39314,
39328,
39342,
39343,
39356,
39358,
39362,
39363,
39366,
39368,
39369,
39372,
39382,
39424,
39456,
39784,
39864,
39945,
41323,
41468,
41472,
41665,
42025,
42336,
42875,
42898,
43264,
43268,
43375,
43688,
43689,
43691,
43692,
43775,
43932,
44375,
44676,
44977,
45056,
45360,
45361,
45362,
45363,
45364,
45365,
45366,
45367,
45368,
45369,
45632,
45684,
45760,
45864,
45873,
45927,
45947,
45957,
45978,
46256,
46368,
46556,
46593,
46608,
46613,
46615,
46619,
46624,
46626,
46630,
46632,
46633,
46635,
46637,
46640,
46641,
46642,
46643,
46644,
46645,
46646,
46647,
46648,
46649,
46650,
46651,
46652,
46653,
46655,
46656,
46657,
46658,
46660,
46661,
46662,
46663,
46664,
46665,
46668,
46672,
46673,
46677,
46684.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 9812 values, from 25 to 999964).
n\r | 0 | 1 |
2 | 5210 | 4602 | 2 |
3 | 4121 | 3166 | 2525 | 3 |
4 | 3254 | 2332 | 1956 | 2270 | 4 |
5 | 1987 | 1784 | 1922 | 2074 | 2045 | 5 |
6 | 2281 | 1536 | 1299 | 1840 | 1630 | 1226 | 6 |
7 | 1906 | 1409 | 1397 | 1282 | 1282 | 1227 | 1309 | 7 |
8 | 2150 | 1320 | 964 | 1055 | 1104 | 1012 | 992 | 1215 | 8 |
9 | 2149 | 1016 | 880 | 1002 | 1196 | 777 | 970 | 954 | 868 | 9 |
10 | 608 | 566 | 1051 | 890 | 1149 | 1379 | 1218 | 871 | 1184 | 896 | 10 |
11 | 987 | 903 | 868 | 877 | 881 | 895 | 888 | 886 | 848 | 924 | 855 |
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.