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%I #40 Jul 14 2019 22:25:47
%S 0,2,12,8,36,16,60,26,84,34,132,46,168,56,180,74,240,82,252,94,324,
%T 106,372,118,420,134,432,142,492,146,540,158,600,166,648,178,660,194,
%U 720,202,780,214,816,226,840,254,912,262,1020,278
%N Minimal difference between the smallest and largest of n consecutive large primes that form a symmetric n-tuplet as permitted by divisibility considerations.
%C For the definition of n-tuplet and minimal differences without the symmetry restriction, see A008407. In particular, a(n) >= A008407(n).
%C 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).
%C Smallest primes starting a shortest symmetric n-tuplet are given in A266512.
%C For odd n, a(n) is divisible by 12.
%H N. Makarova and C. Rivera, <a href="http://www.primepuzzles.net/problems/prob_062.htm">Problem 62. Symmetric k-tuples of consecutive primes</a>.
%e For n=3, any shortest symmetric n-tuplet has the form (p, p+6, p+12) and thus a(3)=12.
%e From _Jon E. Schoenfield_, Jan 05 2016: (Start)
%e 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).
%e 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.
%e For n = 3, 5, ..., 29, the lexicographically first vector (d(1), d(2), ..., d((n-1)/2)) permitted by divisibility considerations is as follows:
%e n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e --+-------------------------------------------------------
%e 3| 6
%e 5| 18 12
%e 7| 30 18 12
%e 9| 42 30 18 12
%e 11| 66 60 36 24 6
%e 13| 84 66 60 36 24 6
%e 15| 90 84 66 60 36 24 6
%e 17|120 108 90 78 60 48 42 18
%e 19|126 120 114 96 84 54 36 30 6
%e 21|162 150 132 120 108 102 78 48 42 18
%e 23|186 180 150 144 126 96 84 66 60 54 30
%e 25|210 186 180 150 144 126 96 84 66 60 54 30
%e 27|216 210 204 180 126 120 114 96 84 54 36 30 6
%e 29|246 216 210 204 186 174 144 126 90 84 66 60 24 6
%e (End)
%Y Cf. A008407, A175309, A266512, A266676.
%K nonn,more
%O 1,2
%A _Max Alekseyev_, Dec 30 2015
%E a(1)-a(10) from _Natalia Makarova_
%E a(11)-a(14), a(16) from _Dmitry Petukhov_
%E a(15) and a(17)-a(18) from Jaroslaw Wroblewski
%E a(20) from _Natalia Makarova_ and Jaroslaw Wroblewski
%E a(19), a(21), a(23), a(25), a(27), a(29) from _Jon E. Schoenfield_, Jan 02 2016, Jan 05 2016
%E a(22), a(24), a(26), a(28), a(30) from _Natalia Makarova_, Jul 06 2016
%E a(31)-a(50) from _Vladimir Chirkov_, Jul 08 2016
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