Possible Permutations With Mod

[Max Alekseyev]

Ferenc,

It seems that you considered numbers (prime(n) mod n)+1 rather than
(prime(n) mod n).
I've extended the search to primes below 10^9 and got that only the
following numbers below 2000 are not represented as prime(n) mod n:

330, 354, 364, 368, 370, 378, 392, 400, 420, 612, 660, 696, 740, 1218,
1240, 1254, 1308, 1320, 1770, 1782, 1914

In particular,
the smallest n satisfying prime(n) mod n=44 (corresponding to "45" in
your list) is n=100371;
the smallest n satisfying prime(n) mod n=110 (corresponding to "111"
in your list) is n=27067317;
etc.

Max

On 3/25/07, Ferenc Adorjan <fadorjan at freemail.hu> wrote:
> Dear Max, Seqfans,
>
> The situation seems rather curious (to me). I do understand that you
> "believe" that  A004648  (prime(n) mod(n)) contains all positive numbers,
> but numerical tests show something different. I also understand, that
> numerics never proves anything. However, I calculated A004648 up to 50000
> terms and found that the numbers below did not show up (least 2000). After
> extending the calculation up to 10^5, still the very same set of numbers
> were missing below 2000.
> Could anyone explain this?
>
> Ferenc
>
> The numbers:
> 45, 111, 127, 163, 166, 181, 187, 199, 209, 211, 213, 223, 331, 337, 355,
> 361, 364, 365, 369, 371, 377, 379, 385, 391, 393, 395, 397, 401, 405, 421,
> 429, 432, 433, 435, 437, 456, 463, 493, 499, 505, 533, 549, 551, 601, 607,
> 613, 617, 639, 643, 659, 661, 673, 687, 693, 697, 707, 723, 727, 731, 735,
> 739, 741, 745, 749, 771, 835, 859, 877, 881, 887, 892, 925, 943, 955, 959,
> 973, 978, 981, 985, 986, 997, 1001, 1009, 1021, 1023, 1027, 1041, 1043,
> 1057, 1063, 1067, 1071, 1077, 1079, 1091, 1123, 1135, 1153, 1156, 1159,
> 1165, 1177, 1181, 1183, 1187, 1189, 1195, 1199, 1201, 1205, 1207, 1213,
> 1219, 1225, 1237, 1239, 1241, 1243, 1245, 1247, 1251, 1255, 1257, 1277,
> 1287, 1289, 1293, 1299, 1305, 1309, 1321, 1327, 1331, 1333, 1336, 1345,
> 1357, 1365, 1369, 1383, 1387, 1393, 1395, 1397, 1409, 1420, 1429, 1453,
> 1465, 1471, 1475, 1516, 1519, 1529, 1531, 1533, 1535, 1537, 1541, 1543,
> 1545, 1547, 1549, 1551, 1553, 1557, 1561, 1567, 1569, 1573, 1579, 1585,
> 1591, 1595, 1603, 1605, 1606, 1607, 1609, 1611, 1613, 1615, 1621, 1627,
> 1633, 1637, 1646, 1647, 1649, 1651, 1653, 1655, 1656, 1657, 1659, 1661,
> 1665, 1669, 1684, 1695, 1699, 1711, 1713, 1715, 1717, 1719, 1721, 1723,
> 1733, 1735, 1737, 1741, 1743, 1747, 1753, 1771, 1783, 1795, 1799, 1801,
> 1805, 1807, 1809, 1813, 1815, 1816, 1817, 1819, 1825, 1829, 1831, 1833,
> 1835, 1837, 1839, 1841, 1843, 1846, 1847, 1849, 1861, 1871, 1873, 1875,
> 1879, 1891, 1893, 1903, 1909, 1913, 1915, 1928, 1933, 1937, 1951, 1957,
> 1959, 1961, 1963, 1967, 1975, 1981, 1983
>
>