A number whose digits are in nondecreasing order. more
The first 600 plaindromes :
1,
2,
3,
4,
5,
6,
7,
8,
9,
11,
12,
13,
14,
15,
16,
17,
18,
19,
22,
23,
24,
25,
26,
27,
28,
29,
33,
34,
35,
36,
37,
38,
39,
44,
45,
46,
47,
48,
49,
55,
56,
57,
58,
59,
66,
67,
68,
69,
77,
78,
79,
88,
89,
99,
111,
112,
113,
114,
115,
116,
117,
118,
119,
122,
123,
124,
125,
126,
127,
128,
129,
133,
134,
135,
136,
137,
138,
139,
144,
145,
146,
147,
148,
149,
155,
156,
157,
158,
159,
166,
167,
168,
169,
177,
178,
179,
188,
189,
199,
222,
223,
224,
225,
226,
227,
228,
229,
233,
234,
235,
236,
237,
238,
239,
244,
245,
246,
247,
248,
249,
255,
256,
257,
258,
259,
266,
267,
268,
269,
277,
278,
279,
288,
289,
299,
333,
334,
335,
336,
337,
338,
339,
344,
345,
346,
347,
348,
349,
355,
356,
357,
358,
359,
366,
367,
368,
369,
377,
378,
379,
388,
389,
399,
444,
445,
446,
447,
448,
449,
455,
456,
457,
458,
459,
466,
467,
468,
469,
477,
478,
479,
488,
489,
499,
555,
556,
557,
558,
559,
566,
567,
568,
569,
577,
578,
579,
588,
589,
599,
666,
667,
668,
669,
677,
678,
679,
688,
689,
699,
777,
778,
779,
788,
789,
799,
888,
889,
899,
999,
1111,
1112,
1113,
1114,
1115,
1116,
1117,
1118,
1119,
1122,
1123,
1124,
1125,
1126,
1127,
1128,
1129,
1133,
1134,
1135,
1136,
1137,
1138,
1139,
1144,
1145,
1146,
1147,
1148,
1149,
1155,
1156,
1157,
1158,
1159,
1166,
1167,
1168,
1169,
1177,
1178,
1179,
1188,
1189,
1199,
1222,
1223,
1224,
1225,
1226,
1227,
1228,
1229,
1233,
1234,
1235,
1236,
1237,
1238,
1239,
1244,
1245,
1246,
1247,
1248,
1249,
1255,
1256,
1257,
1258,
1259,
1266,
1267,
1268,
1269,
1277,
1278,
1279,
1288,
1289,
1299,
1333,
1334,
1335,
1336,
1337,
1338,
1339,
1344,
1345,
1346,
1347,
1348,
1349,
1355,
1356,
1357,
1358,
1359,
1366,
1367,
1368,
1369,
1377,
1378,
1379,
1388,
1389,
1399,
1444,
1445,
1446,
1447,
1448,
1449,
1455,
1456,
1457,
1458,
1459,
1466,
1467,
1468,
1469,
1477,
1478,
1479,
1488,
1489,
1499,
1555,
1556,
1557,
1558,
1559,
1566,
1567,
1568,
1569,
1577,
1578,
1579,
1588,
1589,
1599,
1666,
1667,
1668,
1669,
1677,
1678,
1679,
1688,
1689,
1699,
1777,
1778,
1779,
1788,
1789,
1799,
1888,
1889,
1899,
1999,
2222,
2223,
2224,
2225,
2226,
2227,
2228,
2229,
2233,
2234,
2235,
2236,
2237,
2238,
2239,
2244,
2245,
2246,
2247,
2248,
2249,
2255,
2256,
2257,
2258,
2259,
2266,
2267,
2268,
2269,
2277,
2278,
2279,
2288,
2289,
2299,
2333,
2334,
2335,
2336,
2337,
2338,
2339,
2344,
2345,
2346,
2347,
2348,
2349,
2355,
2356,
2357,
2358,
2359,
2366,
2367,
2368,
2369,
2377,
2378,
2379,
2388,
2389,
2399,
2444,
2445,
2446,
2447,
2448,
2449,
2455,
2456,
2457,
2458,
2459,
2466,
2467,
2468,
2469,
2477,
2478,
2479,
2488,
2489,
2499,
2555,
2556,
2557,
2558,
2559,
2566,
2567,
2568,
2569,
2577,
2578,
2579,
2588,
2589,
2599,
2666,
2667,
2668,
2669,
2677,
2678,
2679,
2688,
2689,
2699,
2777,
2778,
2779,
2788,
2789,
2799,
2888,
2889,
2899,
2999,
3333,
3334,
3335,
3336,
3337,
3338,
3339,
3344,
3345,
3346,
3347,
3348,
3349,
3355,
3356,
3357,
3358,
3359,
3366,
3367,
3368,
3369,
3377,
3378,
3379,
3388,
3389,
3399,
3444,
3445,
3446,
3447,
3448,
3449,
3455,
3456,
3457,
3458,
3459,
3466,
3467,
3468,
3469,
3477,
3478,
3479,
3488,
3489,
3499,
3555,
3556,
3557,
3558,
3559,
3566,
3567,
3568,
3569,
3577,
3578,
3579,
3588,
3589,
3599,
3666,
3667,
3668,
3669,
3677,
3678,
3679,
3688,
3689,
3699,
3777,
3778,
3779,
3788,
3789,
3799,
3888,
3889,
3899,
3999,
4444,
4445,
4446,
4447,
4448,
4449,
4455,
4456,
4457,
4458,
4459,
4466.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1307503 values, from 1 to 999999999999999).
n\r | 0 | 1 |
2 | 361710 | 945793 | 2 |
3 | 435871 | 435816 | 435816 | 3 |
4 | 244750 | 322831 | 116960 | 622962 | 4 |
5 | 11628 | 38775 | 116400 | 320450 | 820250 | 5 |
6 | 120580 | 315251 | 120565 | 315291 | 120565 | 315251 | 6 |
7 | 186784 | 186791 | 186782 | 186782 | 186782 | 186791 | 186791 | 7 |
8 | 157982 | 215119 | 80580 | 233683 | 86768 | 107712 | 36380 | 389279 | 8 |
9 | 145291 | 145272 | 145272 | 145290 | 145272 | 145272 | 145290 | 145272 | 145272 | 9 |
10 | 0 | 15 | 120 | 680 | 3060 | 11628 | 38760 | 116280 | 319770 | 817190 | 10 |
11 | 11439 | 51480 | 123552 | 205920 | 261360 | 261360 | 205920 | 123552 | 51480 | 11440 | 0 |
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.