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good primes
A prime pn such that pn2 > pn-ipn+i for every 0 < i < n. more

The first 600 good primes :
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967, 1009, 1031, 1087, 1151, 1213, 1277, 1367, 1399, 1423, 1427, 1543, 1597, 1847, 1861, 1867, 1871, 1973, 1987, 1993, 1997, 2063, 2203, 2237, 2267, 2333, 2339, 2521, 2531, 2539, 2609, 2647, 2657, 2677, 2683, 2687, 2999, 3163, 3167, 3251, 3299, 3433, 3449, 3457, 3461, 3511, 3527, 3821, 3847, 3907, 3989, 4001, 4201, 4211, 4217, 4229, 4441, 4447, 4481, 4507, 4621, 4637, 4783, 4861, 4871, 4903, 4931, 4967, 5381, 5387, 5407, 5413, 5623, 5639, 5647, 5651, 6007, 6029, 6037, 6043, 6067, 6197, 6257, 6263, 6521, 6547, 6551, 6563, 6637, 6653, 6659, 6689, 6761, 6779, 6823, 6947, 7411, 7451, 7457, 7477, 7517, 7523, 7537, 8039, 8053, 8081, 8087, 8161, 8167, 8209, 8219, 8501, 8513, 8521, 8597, 8623, 8663, 8677, 8689, 8999, 9103, 9127, 9133, 9151, 9277, 9311, 9319, 9337, 9371, 9391, 9397, 9413, 9613, 10007, 10037, 10061, 10067, 10091, 10133, 10589, 10597, 10831, 10847, 10853, 11047, 11057, 11239, 11657, 11677, 11699, 11777, 11807, 11887, 11897, 12197, 12227, 12239, 12373, 12377, 12473, 12487, 12889, 13669, 13679, 13687, 14243, 14293, 14321, 14369, 14387, 14401, 14407, 14519, 14533, 14699, 14713, 15013, 15053, 15073, 15217, 15227, 15259, 15269, 16411, 16417, 16547, 16811, 16823, 16829, 16871, 16879, 16979, 17291, 17317, 17377, 17383, 17783, 17827, 17837, 17881, 17903, 17909, 18899, 18911, 19001, 19139, 19207, 19373, 19379, 19417, 19421, 19889, 19913, 19961, 20681, 20707, 20717, 20743, 20849, 20873, 20897, 20981, 21139, 21313, 21377, 21467, 21481, 21487, 21491, 21557, 22531, 22541, 22613, 22619, 22637, 22691, 22697, 22961, 23003, 23497, 23531, 23537, 23549, 23557, 23741, 24763, 24841, 24907, 24917, 25087, 25111, 25117, 25147, 25301, 25523, 25537, 25561, 25577, 25913, 26669, 26681, 27397, 27407, 27427, 27611, 27673, 27689, 27733, 27737, 28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089, 40423, 40427, 40459, 40471, 40483, 40693, 40697, 40739, 40751, 40759, 40801, 40813, 40819, 41011, 41039, 41113, 41131, 41141, 41177, 41507, 41513, 41579, 41593, 41603, 41843, 41849, 41879, 41887, 41941, 41947, 43573, 43577, 43889, 43933, 43943, 43961, 44449, 44483, 44491, 44497, 45817, 45943, 46021, 46049, 46091, 46133, 46141, 46399, 46439, 46993, 47041, 47051, 47087, 47111, 47119, 47269, 47279, 47287, 47293, 47491, 47497, 48299, 48311, 48337, 48397, 48407, 48463, 48473, 48479, 48731, 48751, 48757, 48973, 48989, 49103, 50741, 50767, 50821, 50833, 51109, 51131, 51193, 51197, 51329, 51341, 51407, 51413, 51419, 52501, 52511, 52529, 52541, 53591, 54251, 54269, 54311, 54319, 54347, 54361, 54367, 54401, 55579, 55603, 55619, 55661, 55763, 55787, 55793, 56359, 56369, 56377, 56431, 56437, 56467, 56473, 58889, 58897, 60589, 60601, 60607, 61211, 61253, 61283, 61331, 61441, 61463, 61469, 62459, 62467, 63241, 63277, 63299, 63311, 64553, 64567, 64577, 64601, 64849, 64871, 64877, 64997, 65027, 65089, 65099, 65353, 65407, 65519, 65537, 66293, 66337, 66343, 66359, 66449, 66457, 66491, 66697, 66841, 66851, 66919, 67033, 67121, 69653, 69677, 69737, 69761, 69809, 69821, 69827, 69991, 69997, 70001, 70099, 70111, 70117, 70823, 70841, 70913, 71143, 71233, 73277, 73291, 73303, 73327, 73351, 73361, 73517, 73523, 73547, 73939, 73999, 74017, 74047, 74131, 74143, 74159, 74687, 74699, 74707, 75133, 75149, 75161, 75167, 76463, 77137, 77237, 77261, 77417, 77471, 77477, 79031, 79087, 79103, 79133, 79147, 79273, 79531, 79549, 79601, 79757, 79769, 79801, 79811, 80141, 80147, 80447, 80471, 80599, 80651, 80669, 80909, 84047, 84053.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 68658 values, from 5 to 199968539).

n\r 0  1 
2068658 2 
303321235446 3 
4034006034652 4 
5117976193731521416094 5 
603321200035446 6 
70103401203311058123181049812411 7 
8017033017340016973017312 8 
9011073118110111841187801095511757 9 
1001797601521401019373016094 10 
1117090747666067162603775936604714762666676

A pictorial representation of the table above
motab
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.