cp's OEIS Frontend

Examples

			For n=3, any shortest symmetric n-tuplet has the form (p, p+6, p+12) and thus a(3)=12.
From _Jon E. Schoenfield_, Jan 05 2016: (Start)
For each n-tuplet (p(1), ..., p(n)) with odd n, let m be its middle prime, i.e., m = p((n+1)/2). Then, since (by symmetry) (p(k) + p(n+1-k))/2 = m for all k = 1..n, we can define the n-tuplet by m and its vector of differences d(j) = m - p(j) for j = 1..(n-1)/2. In other words, given m and d(j) for j = 1..(n-1)/2, the (n-1)/2 primes below m are given by p(j) = m - d(j), and the (n-1)/2 primes above m are given by p(n+1-j) = m + d(j); the difference p(n) - p(1) is thus (m + d(1)) - (m - d(1)) = 2*d(1).
For example, one symmetric 7-tuplet of consecutive primes is (12003179, 12003191, 12003197, 12003209, 12003221, 12003227, 12003239), which can be written as (m-30, m-18, m-12, m, m+12, m+18, m+30) where m=12003209; here we have d(1)=30, d(2)=18, d(3)=12. Among all symmetric 7-tuplets of consecutive primes that satisfy divisibility considerations, the minimal value of d(1) is, in fact, 30, so a(7) = 2*30 = 60.
For n = 3, 5, ..., 29, the lexicographically first vector (d(1), d(2), ..., d((n-1)/2)) permitted by divisibility considerations is as follows:
   n|  1   2   3   4   5   6   7   8   9  10  11  12  13  14
  --+-------------------------------------------------------
   3|  6
   5| 18  12
   7| 30  18  12
   9| 42  30  18  12
  11| 66  60  36  24   6
  13| 84  66  60  36  24   6
  15| 90  84  66  60  36  24   6
  17|120 108  90  78  60  48  42  18
  19|126 120 114  96  84  54  36  30   6
  21|162 150 132 120 108 102  78  48  42  18
  23|186 180 150 144 126  96  84  66  60  54  30
  25|210 186 180 150 144 126  96  84  66  60  54  30
  27|216 210 204 180 126 120 114  96  84  54  36  30   6
  29|246 216 210 204 186 174 144 126  90  84  66  60  24   6
(End)