I have used roman numerals to identify them on this post because unlike the sub-sequences of A186311 which I discussed in June:
- the sequences here are almost certainly not finite and cannot be numbered
- several sequences I have tabulated but have found only the first or first few members
| Sequence | Terms |
| I | 97, 127, 149, 191, 223, 307, 347, 431, 457, 541, 641, 673, 821, 853, 877, 907, 967, 1009, 1031, 1061, 1087, 1151, 1213, 1277, 1319, 1361, 1423, 1447, 1481, 1543, 1597, 1657, 1693, 1777, 1861, 1973, 2081, 2129, 2203, 2237, 2267, 2333, 2371, 2437, 2579, 2609, 2647, 2657, 2707, 2789, 2833, 2953, 2999, 3037, 3079, 3109, 3163, 3181, 3251, 3299, 3449, 3457, 3491, 3527, 3607, 3659, 3671, 3691, 3761, 3793, 3821, 3847, 3877, 3907, 3989, 4001, 4049, 4091, 4127, 4153, 4201, 4211, 4253, 4327, 4441, 4481, 4507, 4547, 4583, 4637, 4721, 4783, 4861, 4931, 4967, 4987, 5077, 5099, 5147, 5167, ... |
| II | 127, 191, 223, 307, 821, 853, 877, 1087, 1151, 1277, 1361, 1423, 1481, 1597, 1657, 1693, 1777, 1861, 1973, 2081, 2129, 2203, 2237, 2267, 2333, 2371, 2999, 3163, 3251, 3299, 3449, 3761, 3907, 3989, 4127, 4201, 4327, 4441, 4507, 4547, 4637, 4721, 4783, 4861, 5147, 5227, 5261, 5381, 5471, 5557, 5623, 5639, 5737, 5779, 6197, 6547, 6761, 6779, 6823, 6857, 6947, 7207, 7283, 7477, 7669, 7867, 7993, 8287, 8419, 8501, 8803, 8923, 8999, 9277, 9337, 9431, 9461, 9587, 9719, 10007, 10061, 10133, 10243, 10427, 10453, 10589, 10831, 10937, 11113, 11239, 11579, 11777, 12037, 12097, 12143, 12197, 12373, 12637, 12889, 13001, ... |
| III | 1087, 1277, 2129, 2333, 2999, 3163, 3251, 3299, 3907, 4637, 4783, 5381, 5557, 5623, 5639, 5779, 6197, 6823, 6947, 8419, 8923, 8999, 9461, 10831, 11239, 11777, 12097, 12143, 12889, 13093, 14143, 14621, 15727, 15859, 16057, 16183, 16411, 17291, 17377, 17657, 18041, 18119, 18899, 19373, 19417, 19661, 20323, 20477, 20849, 21313, 21377, 21467, 21557, 22613, 22853, 23663, 23741, 25219, 25301, 27733, 27941, 28277, 28387, 29009, 29383, 29567, 30631, 31177, 31307, 31379, 31469, 31721, 31957, 32051, 32297, 32561, 32687, 33287, 34123, 34253, 34583, 35023, 35729, 36433, 37781, 37951, 38431, 38447, 38543, 39079, 39293, 39499, 39821, 40343, 40693, 41113, 43391, 43573, 43753, 43777, ... |
| IV | 3299, 4783, 5623, 6947, 8419, 10831, 11777, 12097, 12889, 14143, 15727, 16057, 16183, 18041, 19373, 19661, 21377, 21467, 22613, 23663, 25301, 27733, 28277, 29009, 31469, 31957, 32297, 32687, 34123, 34583, 35729, 36433, 37951, 38431, 43391, 43777, 44249, 44351, 45119, 45817, 45943, 46807, 47491, 47777, 48731, 49919, 52361, 54401, 55201, 58889, 59333, 60077, 60589, 61331, 62969, 68207, 68863, 69991, 72859, 74687, 75011, 78779, 79757, 80599, 80897, 81509, 81611, 82003, 82129, 82457, 85991, 86923, 88643, 89753, 91367, 92551, 93047, 93229, 93463, 95063, 95701, 97367, 100907, 101267, 102761, 103951, 105319, 106087, 106853, 108863, 110419, 111091, 113327, 113891, 114547, 116789, 117497, 118033, 119027, 124337, ... |
| V | 12889, 15727, 22613, 27733, 28277, 31469, 32297, 34123, 37951, 43391, 45817, 45943, 48731, 58889, 59333, 60077, 60589, 68863, 78779, 81509, 82129, 86923, 89753, 91367, 93463, 95701, 97367, 100907, 108863, 117497, 124427, 126001, 134581, 136943, 153059, 153247, 156007, 156217, 159157, 162209, 163973, 165701, 167593, 167861, 176299, 177883, 179897, 186551, 188107, 191413, 191441, 197887, 198811, 200569, 206879, 211859, 212777, 217307, 218357, 220123, 220663, 221159, 221393, 223207, 223823, 225217, 225341, 227363, 236867, 237137, 246889, 252139, 258101, 259099, 263489, 265703, 275129, 276817, 279109, 281509, 282559, 282661, 285937, 288647, 289717, 289837, 290317, 290593, 293599, 296909, 297581, 298153, 302551, 305717, 307243, 307817, 311533, 312197, 317957, 321289, ... |
| VI | 12889, 15727, 28277, 31469, 34123, 48731, 58889, 60589, 95701, 117497, 153059, 156007, 156217, 163973, 167593, 167861, 191413, 198811, 218357, 223207, 227363, 236867, 237137, 246889, 265703, 281509, 282661, 285937, 288647, 307243, 311533, 323333, 332393, 338119, 342037, 360163, 360749, 370373, 371027, 372607, 404941, 422537, 425779, 428137, 449543, 458309, 459593, 464741, 492227, 492377, 507289, 529927, 530501, 531793, 531977, 535481, 542683, 552469, 561907, 566521, 572023, 595927, 607301, 617647, 618799, 633739, 646519, 663517, 673747, 678719, 698239, 703631, 724721, 736357, 761347, 769541, 779327, 780799, 787427, 795643, 809701, 822517, 828349, 832583, 838349, 840439, 844421, 844601, 856693, 860239, 863479, 880661, 886741, 886967, 889871, 928769, 947707, 949381, 954827, 959449, ... |
| VII | 34123, 58889, 198811, 218357, 223207, 236867, 246889, 265703, 281509, 323333, 338119, 360163, 360749, 370373, 404941, 566521, 595927, 633739, 678719, 724721, 736357, 780799, 787427, 809701, 822517, 832583, 838349, 844421, 860239, 886967, 949381, 1047647, 1050713, 1053691, 1053953, 1078331, 1102663, 1124951, 1128889, 1136557, 1138363, 1140091, 1146661, 1155821, 1183687, 1195669, 1210351, 1242739, 1243093, 1246181, 1280101, 1307051, 1313567, 1324567, 1349651, 1352749, 1357333, 1388587, 1397437, 1440203, 1444411, 1461283, 1463447, 1464241, 1468387, 1491377, 1505083, 1531987, 1562051, 1568867, 1570837, 1573477, 1576483, 1599803, 1602817, 1629541, 1648181, 1671907, 1678837, 1775171, 1802597, 1834597, 1876949, 1889957, 1894577, 1895479, 1904407, 1910257, 1942723, 1966787, 2010881, 2022971, 2045311, 2121121, 2127269, 2153051, 2192549, 2208053, 2238931, 2242651, ... |
| VIII | 223207, 281509, 338119, 360163, 370373, 404941, 633739, 678719, 724721, 736357, 809701, 822517, 832583, 838349, 844421, 860239, 949381, 1053953, 1078331, 1102663, 1124951, 1138363, 1140091, 1155821, 1195669, 1210351, 1242739, 1280101, 1307051, 1313567, 1349651, 1388587, 1444411, 1464241, 1562051, 1599803, 1678837, 1775171, 1834597, 1889957, 1895479, 1966787, 2010881, 2045311, 2127269, 2238931, 2305271, 2527381, 2598091, 2637911, 2784473, 2867213, 2995973, 3056327, 3112237, 3117421, 3126653, 3135017, 3269639, 3376937, 3393311, 3501587, 3579481, 3593311, 3595589, 3654317, 3780017, 3789701, 3829079, 3851587, 3889157, 3917411, 3933731, 4024667, 4087597, 4165927, 4271557, 4289599, 4356661, 4535819, 4640717, 4652507, 4738777, 4876607, 4975757, 4998541, 5103029, 5122631, 5211209, 5274551, 5451241, 5589127, 5651707, 5724097, 5789857, 5845309, 5888873, 6005849, 6034393, 6082561, ... |
| IX | 338119, 370373, 736357, 822517, 838349, 1102663, 1138363, 1444411, 1464241, 1562051, 1599803, 1834597, 1895479, 1966787, 2010881, 2127269, 2238931, 2305271, 2598091, 2784473, 2867213, 2995973, 3056327, 3112237, 3117421, 3135017, 3376937, 3393311, 3593311, 3595589, 3829079, 4024667, 4087597, 4165927, 4640717, 4652507, 4738777, 4975757, 5274551, 5451241, 5589127, 5724097, 5789857, 5888873, 6005849, 6034393, 6085103, 6371537, 6655531, 6756817, 7118077, 7743371, 7792297, 7803611, 7932781, 7970471, 8001491, 8010109, 8027807, 8211331, 8380037, 8546381, 8640679, 8655611, 8786359, 8917663, 8996521, 9108419, 9111449, 9145033, 9285161, 9286799, 9535573, 9943663, 10001203, 10156187, 10253149, 10270591, 10289329, 10509461, 10580147, 10634587, 10823149, 10839061, 10862557, 10864093, 11193167, 11262227, 11276687, 11296801, 11502019, 11581711, 11724127, 11883601, 11900501, 11928299, 11951699, 12009329, 12183877, 12667777, ... |
| X | 370373, 736357, 838349, 1138363, 1464241, 1562051, 1895479, 1966787, 2010881, 2127269, 2238931, 2305271, 2598091, 2784473, 3056327, 3112237, 3117421, 3595589, 3829079, 4024667, 4640717, 4738777, 5274551, 5451241, 6085103, 6371537, 6812339, 7118077, 7743371, 8027807, 8917663, 9108419, 9111449, 9285161, 10156187, 10253149, 10509461, 10580147, 10864093, 11193167, 11581711, 11724127, 12703501, 12819329, 12833237, 13211059, 13416457, 13738849, 14247113, 14848511, 15020857, 15423427, 15825871, 15834587, 15856201, 16012637, 16377589, 16432243, 16684471, 16873867, 18347759, 18852077, 19227907, 19288631, 19294523, 19428313, 19716649, 20024483, 20253347, 20285263, 20388727, 21500429, 21782647, 21791477, 21919003, ... |
| XI | 370373, 838349, 1562051, 2010881, 2598091, 2784473, 3117421, 3595589, 3829079, 4640717, 6085103, 7743371, 8917663, 9108419, 11193167, 11581711, 13738849, 15020857, 15834587, 16684471, 19294523, 19428313, 20253347, 20388727, 21500429, 21919003, 22337291, 22539899, 23008549, 23764519, 24120379, 24138187, 24416591, 25106509, 25703651, 26662591, 27915907, 27980987, ... |
| XII | 370373, 3829079, 8917663, 11193167, 13738849, 15834587, 19294523, 19428313, 20253347, 20388727, 21500429, 21919003, 22337291, 22539899, 23008549, 23764519, 24120379, 24416591, 25106509... |
| XIII | 8917663, 11193167, 13738849, 15834587, 20388727, 22539899, 23008549, 24120379, ... |
| XIV | 11, 97, 127, 1087, 3299, 12889, 12889, 34123, 223207, 338119, 370373, 370373, 370373, 8917663, 20388727, 20388727, ... |
Odd Primes for Which the (n+1) Closest Primes Are Larger
Excluding the final sequence, the sequences above are odd primes for which the (n+1) closest primes are larger.Sequence XIV is the smallest odd prime for which the n closest primes are larger.
What is surprising about the OEIS page is that there exist sequences for which the two, three, four, five, six, seven, and eight closest primes are smaller, but the sequences tabulated above are not on OEIS.
The only explanation I can think of is that whilst there is a smallest prime (2), ever since Euclid it has been known that there is no largest prime. This means that smallest prime 2 will always be the smallest prime for which the n closest primes are larger, regardless of how large n is. Plainly, every other prime is larger than 2. However, one can of course correct this by limiting the sequence to odd primes to create the fourteen tabulated sequences above.
Although I have not made a detailed study, the sequences above appear extremely similar in size to those noted above as present on OEIS A075051.
A075051 does suggest it would be hard work extending sequence XIV beyond the smallest odd prime for which the sixteen closest primes are larger — it could be more than an order of magnitude larger than 20,388,727.
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