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%I #55 Dec 05 2023 12:05:28
%S 5,18731,683783,98303927,60335249959,1169769749219,3945769040699039,
%T 159067808851610657
%N Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.
%C Least n-tuply balanced primes: primes which are averages of both their immediate neighbors, their second neighbors, their third neighbors, ... and their n-th neighbors.
%C a(9) <= 6919940122097246597. The solution was found by the BOINC project "SPT test project". - _Natalia Makarova_, Nov 25 2023
%H Stop@home, <a href="http://stop.inferia.ru/">BOINC project</a> to search all up to 2^64. [Dead link]
%H Symmetric Prime Tuples, <a href="https://boinc.termit.me/adsl/">SPT test project</a>.
%F a(n) = A151800^(n)(A175309(2n)), i.e., A151800 applied n times on A175309(2n). - _Max Alekseyev_, Jul 26 2014
%e In 5-tuple of consecutive primes (18713, 18719, 18731, 18743, 18749), the primes are symmetric w.r.t. its central prime 18731, since 18713+18749 = 18719+18743 = 2*18731, and this is the smallest such 5-tuple. Hence, a(2)=18731.
%e Alternatively, the symmetry can be seen from the differences between consecutive primes. For (18713, 18719, 18731, 18743, 18749), the differences are (6,12,12,6).
%t Table[i = n + 2;
%t While[x = Differences[Table[Prime[k + i], {k, -n, n}]];
%t x != Reverse[x], i++]; Prime[i], {n, 3}] (* _Robert Price_, Oct 12 2019 *)
%Y Cf. A001223, A055381, A055382, A006562, A051795, A081415, A096710.
%K more,nonn
%O 1,1
%A _Jud McCranie_, Jun 23 2000
%E a(6) from _Donovan Johnson_, Mar 09 2008
%E Definition corrected by _Max Alekseyev_, Jul 29 2014
%E a(7) from _Dmitry Petukhov_, added by _Max Alekseyev_, Nov 03 2014
%E a(8) from SPT project, added by _Dmitry Petukhov_, Apr 06 2017
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