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%I #6 Sep 08 2022 08:45:30
%S 156241,253969,674071,1127629,1285981,1372543,1406683,1464751,1471573,
%T 1479871,1708351,1739833,1829203,2056381,2233123,2822923,2854933,
%U 2970109,3369193,3494923,3534913,3633139,3771583,3903643,4129381
%N Balanced primes p of the form (r+q+s-1)/2, where r, q, s are consecutive primes and q is a balanced prime.
%C The primes q arising here are in A129241.
%C Subsequence of A129191, where q need not be balanced.
%e 253969 = (169307+169313+169319-1)/2 = A006562(937) is a balanced prime, it has distance 18 to the preceding prime 253951 and to the next prime 253987. 169307, 169313, 169319 are consecutive primes and 169313 = A006562(666) is a balanced prime (distance 6), hence 253969 is a term.
%t Select[Select[(Total[#]-1)/2&/@(Select[Partition[Prime[ Range[ 500000]], 3,1],Last[#]-#[[2]]==#[[2]]-First[#]&]), PrimeQ], NextPrime[#]-# == #-NextPrime[#,-1]&] (* _Harvey P. Dale_, May 11 2011 *)
%o (Magma) [ p: q in PrimesInInterval(3, 2900000) | r+s eq 2*q and IsPrime(p) and PreviousPrime(p)+NextPrime(p) eq 2*p where p is (r+q+s-1) div 2 where r is PreviousPrime(q) where s is NextPrime(q) ];
%Y Cf. A006562 (balanced primes), A129191, A129241.
%K nonn
%O 1,1
%A _Klaus Brockhaus_, Apr 05 2007
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