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A126554 Arithmetic mean of two consecutive balanced primes (of order one).

%I A126554 #17 Mar 28 2020 04:28:18
%S A126554 29,105,165,192,234,260,318,468,578,600,630,693,840,962,1040,1113,
%T A126554 1155,1205,1295,1439,1629,1750,1830,2097,2352,2547,2790,2933,3135,
%U A126554 3310,3475,3685,3873,4211,4433,4527,4627,4674,4842,5050,5110,5208,5345,5390,5478
%N A126554 Arithmetic mean of two consecutive balanced primes (of order one).
%C A126554 Might be called interprimes of order two, since the arithmetic means of two consecutive odd primes (A024675) sometimes are called interprimes.
%C A126554 Balanced primes of order two (A082077) and doubly balanced primes (A051795) have different definitions.
%C A126554 For primes in this sequence (prime interprimes of order two) see A126555.
%H A126554 Muniru A Asiru and Amiram Eldar, <a href="/A126554/b126554.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..5000 from Muniru A Asiru)
%F A126554 a(n) = (A006562(n+1)+A006562(n))/2.
%t A126554 b = {}; a = {}; Do[If[PrimeQ[((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2], AppendTo[a, ((Prime[n + 2] + Prime[n + 1])/2 + (Prime[n + 1] + Prime[n])/2)/2]], {n, 1, 1000}]; Do[AppendTo[b, (a[[k + 1]] + a[[k]])/2], {k, 1, Length[a] - 1}]; b
%o A126554 (PARI) {m=6000;a=0;p=2;q=3;r=5;while(r<=m,if((p+r)/2==q,if(a>0,print1((a+q)/2,","));a=q);p=q;q=r;r=nextprime(r+1))} \\ _Klaus Brockhaus_, Jan 05 2007
%o A126554 (GAP) P:=Filtered([1..6000],IsPrime);;P1:=List(Filtered(List([0..Length(P)-3],k->List([1..3],j->P[j+k])),i->Sum(i)/3=i[2]),m->m[2]);;
%o A126554 a:=List([1..Length(P1)-1],n->(P1[n+1]+P1[n])/2); # _Muniru A Asiru_, Mar 31 2018
%Y A126554 Cf. A006562, A024675, A082077, A051795, A126555.
%K A126554 nonn
%O A126554 1,1
%A A126554 _Artur Jasinski_, Dec 27 2006
%E A126554 Edited by _Klaus Brockhaus_, Jan 05 2007