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%I #21 Apr 15 2018 13:48:36
%S 683783,1056317,1100261,2241709,2815301,4746359,10009049,12003209,
%T 13810981,14907649,15403009,15730067,16595081,17518201,19755301,
%U 20378327,21006487,21574453,21579983,22237121,22625179,25876901
%N Triply balanced primes: primes which are averages of both their immediate neighbor, their second neighbors and their third neighbors.
%C Equivalently, primes which are balanced primes of orders 1, 2, and 3. - _Muniru A Asiru_, Apr 08 2018
%H Jud McCranie, <a href="/A081415/b081415.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Balanced_prime">Balanced prime</a>
%e p = 683383: 683747 + ... + p + .. + 683819 = 7p; 683759 + .. + p + .. + 683807 = 5p; 683777 + p + 683789 = 3p.
%t a = {}; Do[p = 2Prime[n]; If[p == Prime[n - 1] + Prime[n + 1], If[p == Prime[n - 2] + Prime[n + 2], If[p == Prime[n - 3] + Prime[n + 3], {n, 5, 1100000}] (* _Robert G. Wilson v_, Jun 28 2004 *)
%t Transpose[Select[Partition[Prime[Range[1620000]],7,1],(#[[1]]+#[[7]])/2 == (#[[2]]+#[[6]])/2==(#[[3]]+#[[5]])/2==#[[4]]&]][[4]] (* _Harvey P. Dale_, Sep 13 2013 *)
%o (GAP) P:=Filtered([1,3..3*10^7+1],IsPrime);;
%o a:=Intersection(List([1,2,3],b->List(Filtered(List([0..Length(P)-(2*b+1)],k->List([1..2*b+1],j->P[j+k])),i->Sum(i)/(2*b+1)=i[b+1]),m->m[b+1]))); # _Muniru A Asiru_, Apr 08 2018
%Y Cf. A006562, A051795, A055380, A096710.
%K nonn
%O 1,1
%A _Labos Elemer_, Apr 02 2003
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