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balanced primes
A prime which is the average of the two surrounding primes. more

The first 600 balanced primes :
5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393, 5563, 5807, 6073, 6263, 6317, 6323, 6367, 6373, 6863, 6977, 7523, 7583, 7823, 7841, 8117, 8713, 8747, 9397, 9473, 9871, 10253, 10607, 10657, 10853, 11299, 11411, 11497, 11731, 11807, 11903, 11933, 12497, 12547, 12583, 12647, 12653, 12841, 12973, 13043, 13177, 13457, 13463, 14543, 14747, 14753, 15161, 15193, 15313, 15467, 15767, 15797, 15803, 15907, 15913, 16097, 16223, 16427, 16487, 16567, 16619, 16787, 16937, 16987, 17047, 17327, 17431, 17477, 17483, 17851, 18217, 18223, 18341, 18433, 18731, 19463, 19477, 19483, 19577, 19603, 19739, 19949, 20107, 20123, 20161, 20201, 20347, 20521, 20731, 21163, 21661, 21911, 21997, 22051, 22073, 22259, 22447, 23327, 23333, 23767, 23801, 23893, 24077, 24097, 24413, 24677, 25391, 25463, 25621, 25673, 26041, 26177, 26183, 26393, 26693, 26723, 26987, 27067, 27773, 28069, 28597, 28927, 29173, 29269, 29333, 29599, 29873, 30059, 30097, 30103, 30313, 30449, 30637, 30643, 30977, 31051, 31327, 32083, 32479, 32497, 32993, 33113, 33857, 34171, 34267, 34543, 34631, 34673, 34877, 35129, 35317, 35753, 35851, 35911, 36067, 36229, 36313, 36677, 36821, 37957, 38183, 38867, 39113, 39863, 40063, 40093, 40111, 40283, 40471, 40493, 40933, 40961, 41017, 41263, 41479, 41513, 41603, 41947, 42187, 42209, 42397, 42533, 42683, 43189, 44041, 44519, 44543, 44741, 45433, 45497, 45757, 46147, 46889, 47303, 47507, 47527, 47869, 48023, 48259, 48533, 48661, 49069, 49633, 50153, 50417, 50441, 51413, 51461, 51551, 52529, 52727, 52957, 53617, 53623, 53939, 54151, 54443, 54623, 54673, 54773, 54851, 55207, 55793, 55843, 56003, 56087, 56093, 56149, 56437, 56773, 56957, 57119, 57853, 58067, 58129, 58237, 58543, 58573, 58693, 58979, 59113, 59393, 60083, 60133, 60337, 60509, 60617, 61553, 61637, 61723, 62207, 62213, 62597, 62633, 63073, 63493, 63527, 63559, 63697, 63703, 63907, 64013, 64627, 65413, 65557, 65707, 65957, 66431, 66809, 66883, 67169, 67511, 67619, 67783, 68099, 68483, 68699, 68743, 69067, 69623, 71011, 71347, 71353, 71843, 71993, 72031, 72481, 72733, 73127, 73477, 73523, 74317, 74471, 74489, 74567, 74897, 75181, 75217, 75527, 75533, 75577, 75641, 75787, 76487, 76561, 76579, 76673, 76913, 77023, 77081, 77557, 77563, 77951, 78173, 78259, 78577, 78797, 78803, 79187, 79411, 79823, 79867, 79973, 80251, 80329, 80917, 80923, 81001, 81077, 81439, 81689, 81749, 81937, 82463, 82493, 82787, 82793, 83003, 83437, 83443, 84053, 84137, 84437, 84443, 85049, 85121, 85837, 86137, 86263, 86981, 87427, 87517, 87547, 88867, 89003, 89107, 89113, 89387, 89393, 89527, 89983, 90511, 90703, 91243, 91297, 91303, 91387, 92003, 92227, 92363, 92581, 93053, 93083, 93377, 94427, 94433, 95267, 95273, 95707, 95917, 95923, 96001, 96377, 96431, 96763, 97441, 97967, 98473, 98597, 99823, 100937, 101273, 101287, 101483, 103177, 103567, 103669, 104003, 104021, 104053, 104113, 104161, 104207, 104543, 104717, 104723, 104953, 105367, 105373, 105607, 105613, 105977, 106297, 106681, 107581, 107693, 108217, 108439, 108637, 108643, 108739, 108923, 109897, 110459, 110603, 110681, 110813, 110933, 111509, 111521, 111611, 111773, 111863, 112181, 112297, 112583, 112939, 113117, 113921, 113963, 114001, 114083, 114473, 114901, 115337, 115757, 115763, 116107, 116719, 116797, 117437, 117757, 118831, 119033, 119267, 119557, 119563, 119759, 120097, 120557, 120563, 120817, 120823, 121007, 121013, 121327, 121421, 121447, 121553, 122099, 122273, 122579, 122833, 123433, 123887, 124459, 125107, 125441, 125539, 125737, 126211, 126317, 126839, 127487, 127643, 127663, 127733, 127843, 128153, 128449, 128467, 128483, 129049, 129209, 129763, 130343, 130483, 130687, 130693, 130829, 131041, 131933, 132511, 132911, 132929, 133103, 133571, 133717, 134053, 134213, 134263, 134333, 134639, 135089, 135301, 135607, 135893, 136733, 137131, 137321, 137393, 137849, 138107, 139079, 139703, 139753, 140351, 140527, 140983, 141067, 141073, 141263, 142019, 142111, 142217, 142369, 142427, 142553, 143159, 143483, 143699, 144247, 144253, 144439, 144583, 144611, 144757, 144967, 145037, 145121, 145213, 145589, 145661, 145879, 146197, 146701.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 10801518 values, from 5 to 9999997543).

n\r 0  1 
2010801518 2 
3053998725401646 3 
40540073005400788 4 
512429250297257929711622428526 5 
6053998720005401646 6 
70151549717907132096795209500917870891516415 7 
802702288026996370269844202701151 8 
9017970371801879018003961800096018024391799671 9 
100242925002971162010297257902428526 10 
11010258721225106119126011659087952107955031164870118905812245221024209

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.