
COMMENTS

The average number of balanced primes, p_n, seems to reach a maximum at the 85th prime, 439, of 32 balanced primes.
A096711(n)/A006880(n) begins 25%, 32%, 33.93%, 25.06%, 20.74%


MATHEMATICA

f[n_] := f[n] = Block[{c = 0, k = 1, p = Prime[n], s = Plus @@ Table[Prime[i], {i, n  1, n + 1}]}, While[k != n  1, If[s == (2k + 1)p, Return[1]]; k++; s = s + Prime[n  k] + Prime[n + k]]; 0]; f[1] = 0; Do[ f[n], {n, 10000}]; s = Prime[ Select[ Range[ 10000], f[ # ] == 1 &]]; Table[ Length[ Select[s, # < 10^n &]], {n, 5}]
