A number n whose digits go up and then go down. more
The first 600 mountain numbers :
120,
121,
130,
131,
132,
140,
141,
142,
143,
150,
151,
152,
153,
154,
160,
161,
162,
163,
164,
165,
170,
171,
172,
173,
174,
175,
176,
180,
181,
182,
183,
184,
185,
186,
187,
190,
191,
192,
193,
194,
195,
196,
197,
198,
230,
231,
232,
240,
241,
242,
243,
250,
251,
252,
253,
254,
260,
261,
262,
263,
264,
265,
270,
271,
272,
273,
274,
275,
276,
280,
281,
282,
283,
284,
285,
286,
287,
290,
291,
292,
293,
294,
295,
296,
297,
298,
340,
341,
342,
343,
350,
351,
352,
353,
354,
360,
361,
362,
363,
364,
365,
370,
371,
372,
373,
374,
375,
376,
380,
381,
382,
383,
384,
385,
386,
387,
390,
391,
392,
393,
394,
395,
396,
397,
398,
450,
451,
452,
453,
454,
460,
461,
462,
463,
464,
465,
470,
471,
472,
473,
474,
475,
476,
480,
481,
482,
483,
484,
485,
486,
487,
490,
491,
492,
493,
494,
495,
496,
497,
498,
560,
561,
562,
563,
564,
565,
570,
571,
572,
573,
574,
575,
576,
580,
581,
582,
583,
584,
585,
586,
587,
590,
591,
592,
593,
594,
595,
596,
597,
598,
670,
671,
672,
673,
674,
675,
676,
680,
681,
682,
683,
684,
685,
686,
687,
690,
691,
692,
693,
694,
695,
696,
697,
698,
780,
781,
782,
783,
784,
785,
786,
787,
790,
791,
792,
793,
794,
795,
796,
797,
798,
890,
891,
892,
893,
894,
895,
896,
897,
898,
1210,
1230,
1231,
1232,
1240,
1241,
1242,
1243,
1250,
1251,
1252,
1253,
1254,
1260,
1261,
1262,
1263,
1264,
1265,
1270,
1271,
1272,
1273,
1274,
1275,
1276,
1280,
1281,
1282,
1283,
1284,
1285,
1286,
1287,
1290,
1291,
1292,
1293,
1294,
1295,
1296,
1297,
1298,
1310,
1320,
1321,
1340,
1341,
1342,
1343,
1350,
1351,
1352,
1353,
1354,
1360,
1361,
1362,
1363,
1364,
1365,
1370,
1371,
1372,
1373,
1374,
1375,
1376,
1380,
1381,
1382,
1383,
1384,
1385,
1386,
1387,
1390,
1391,
1392,
1393,
1394,
1395,
1396,
1397,
1398,
1410,
1420,
1421,
1430,
1431,
1432,
1450,
1451,
1452,
1453,
1454,
1460,
1461,
1462,
1463,
1464,
1465,
1470,
1471,
1472,
1473,
1474,
1475,
1476,
1480,
1481,
1482,
1483,
1484,
1485,
1486,
1487,
1490,
1491,
1492,
1493,
1494,
1495,
1496,
1497,
1498,
1510,
1520,
1521,
1530,
1531,
1532,
1540,
1541,
1542,
1543,
1560,
1561,
1562,
1563,
1564,
1565,
1570,
1571,
1572,
1573,
1574,
1575,
1576,
1580,
1581,
1582,
1583,
1584,
1585,
1586,
1587,
1590,
1591,
1592,
1593,
1594,
1595,
1596,
1597,
1598,
1610,
1620,
1621,
1630,
1631,
1632,
1640,
1641,
1642,
1643,
1650,
1651,
1652,
1653,
1654,
1670,
1671,
1672,
1673,
1674,
1675,
1676,
1680,
1681,
1682,
1683,
1684,
1685,
1686,
1687,
1690,
1691,
1692,
1693,
1694,
1695,
1696,
1697,
1698,
1710,
1720,
1721,
1730,
1731,
1732,
1740,
1741,
1742,
1743,
1750,
1751,
1752,
1753,
1754,
1760,
1761,
1762,
1763,
1764,
1765,
1780,
1781,
1782,
1783,
1784,
1785,
1786,
1787,
1790,
1791,
1792,
1793,
1794,
1795,
1796,
1797,
1798,
1810,
1820,
1821,
1830,
1831,
1832,
1840,
1841,
1842,
1843,
1850,
1851,
1852,
1853,
1854,
1860,
1861,
1862,
1863,
1864,
1865,
1870,
1871,
1872,
1873,
1874,
1875,
1876,
1890,
1891,
1892,
1893,
1894,
1895,
1896,
1897,
1898,
1910,
1920,
1921,
1930,
1931,
1932,
1940,
1941,
1942,
1943,
1950,
1951,
1952,
1953,
1954,
1960,
1961,
1962,
1963,
1964,
1965,
1970,
1971,
1972,
1973,
1974,
1975,
1976,
1980,
1981,
1982,
1983,
1984,
1985,
1986,
1987,
2310,
2320,
2321,
2340,
2341,
2342,
2343,
2350,
2351,
2352,
2353,
2354,
2360,
2361,
2362,
2363,
2364,
2365,
2370,
2371,
2372,
2373,
2374,
2375,
2376,
2380,
2381,
2382,
2383,
2384,
2385,
2386,
2387,
2390,
2391,
2392,
2393,
2394,
2395,
2396,
2397,
2398,
2410,
2420.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 173083 values, from 120 to 456789876543210).
n\r | 0 | 1 |
2 | 115465 | 57618 | 2 |
3 | 57722 | 57701 | 57660 | 3 |
4 | 46124 | 34565 | 69341 | 23053 | 4 |
5 | 89437 | 44756 | 22353 | 11112 | 5425 | 5 |
6 | 38508 | 19220 | 38476 | 19214 | 38481 | 19184 | 6 |
7 | 24727 | 24730 | 24724 | 24720 | 24727 | 24728 | 24727 | 7 |
8 | 27122 | 19663 | 39423 | 9501 | 19002 | 14902 | 29918 | 13552 | 8 |
9 | 19240 | 19233 | 19222 | 19242 | 19232 | 19220 | 19240 | 19236 | 19218 | 9 |
10 | 86732 | 43419 | 21716 | 10857 | 5425 | 2705 | 1337 | 637 | 255 | 0 | 10 |
11 | 30981 | 28195 | 21093 | 12762 | 6156 | 2742 | 2784 | 6246 | 12837 | 21108 | 28179 |
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.