Varieties of Balanced Primes

Recently I passed the 24877 mark. 24877 is a prime number but it's also a balanced prime of order seven, meaning that it is the average of the seven primes preceding it and seven primes succeeding it. The first example of such a prime is 29 because:
$$\begin{gathered} \mathbf{\text{29}}=\frac{\stackrel{\text{seven primes below}}{\overbrace{5+7+11+13+17+19+23}}}{15} \\ +\frac{\mathbf{\text{29}}+\stackrel{\text{seven primes above}}{\overbrace{31+37+41+43+47+53+59}}}{15} \end{gathered}$$Balanced primes range from order one upwards with the first balanced prime of order one being 5 where:$$\mathbf{\text{5}}=\frac{3+\mathbf{\text{5}}+7}{3}$$A subset of the balanced primes are those primes that are doubly balanced, meaning they are averages of both their immediate and their second neighbours. The first example of such a prime is 18731 with surrounding primes of 18713, 18719 below and 18743, 18749 above. We find that:$$\begin{array}{rl}\mathbf{\text{18731}}& =\frac{18719+\mathbf{\text{18731}}+18743}{3}\\ & =\frac{18713+18719+\mathbf{\text{18731}}+18743+18749}{5}\end{array}$$Primes can be triply balanced (the first of these is 683783), quadruply balanced (the first of these is 98303927) and so on. In terms of the counting of the days of our lives, it is only the first five of the doubly balanced primes (18731, 25621, 28069, 30059 and 31051) that we are likely to encounter.

A prime can be balanced in more ways than one but the orders may not be continuous. A doubly balanced prime like 18713 is a balanced prime of order 1 and order 2. Here the orders, 1 and 2, are continuous. However, a prime like 263 can be written as (257 + 263 + 269)/3 but also (179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229 + 233 + 239 + 241 + 251 + 257 + 263 + 269 + 271 + 277 + 281 + 283 + 293 + 307 + 311 + 313 + 317 + 331 + 337 + 347 + 349 + 353)/31. It is a balanced prime of order 1 and order 31, but 1 and 31 are not continuous. Both 263 and 18713 are said to be balanced primes of index 2. The balanced primes of index 1 are those that are balanced in one way only, the first of these being 5.

Unlike the doubly balanced primes (18731, 25621, 28069, 30059, 31051 etc.), the balanced primes of index 2 are relatively frequent. Here is a list of the first of them:

211, 263, 349, 397, 409, 439, 709, 751, 787, 827, 1153, 1187, 1259, 1487, 1523, 1531, 2281, 2287, 2347, 2621, 3037, 3109, 3313, 3329, 3539, 3673, 4357, 4397, 4493, 4951, 4969, 4987, 5189, 5303, 5347, 5857, 6323, 6337, 7583, 7907, 7933, 8429, 8713, 8821

After that, they naturally become less frequent. Here are the first balanced primes of index 3:

53, 607, 977, 1289, 2083, 2351, 4013, 5563, 8803, 10657, 11117, 12583, 14747, 16433, 18731, 22067, 22699, 28477, 32833, 39227, 39749, 41957, 44357, 46229, 46643, 50053, 50123, 51869, 53617, 54469, 56167, 63377, 63527, 66797, 74729, 75217

Even the first few balanced primes of index 4 (157, 353, 8233, 23893, 26183 and 30197) will occur within the lives of most individuals.