A266511 Minimal difference between the smallest and largest of n consecutive large primes that form a symmetric n-tuplet as permitted by divisibility considerations.

0, 2, 12, 8, 36, 16, 60, 26, 84, 34, 132, 46, 168, 56, 180, 74, 240, 82, 252, 94, 324, 106, 372, 118, 420, 134, 432, 142, 492, 146, 540, 158, 600, 166, 648, 178, 660, 194, 720, 202, 780, 214, 816, 226, 840, 254, 912, 262, 1020, 278

Offset: 1

Author

Max Alekseyev, Dec 30 2015

Keywords

• nonn
• more

Comments

For the definition of n-tuplet and minimal differences without the symmetry restriction, see A008407. In particular, a(n) >= A008407(n).

An n-tuplet (p(1),...,p(n)) is symmetric if p(k) + p(n+1-k) is the same for all k=1,2,...,n (cf. A175309).

Smallest primes starting a shortest symmetric n-tuplet are given in A266512.

For odd n, a(n) is divisible by 12.

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)
		

Links

• N. Makarova and C. Rivera, Problem 62. Symmetric k-tuples of consecutive primes.

Crossrefs

Cf. A008407, A175309, A266512, A266676.

Extensions

a(1)-a(10) from Natalia Makarova

a(11)-a(14), a(16) from Dmitry Petukhov

a(15) and a(17)-a(18) from Jaroslaw Wroblewski

a(20) from Natalia Makarova and Jaroslaw Wroblewski

a(19), a(21), a(23), a(25), a(27), a(29) from Jon E. Schoenfield, Jan 02 2016, Jan 05 2016

a(22), a(24), a(26), a(28), a(30) from Natalia Makarova, Jul 06 2016

a(31)-a(50) from Vladimir Chirkov, Jul 08 2016