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It has been conjectured that A004767(n) = 4n+3 is either prime or the mean of two primes in sequence A004767.
See A244952 and A245205 for record values and indices of these records. Some of these records are a(97)=27, a(1139)=120, a(10181)=225, a(93124)=435, a(864901)=894, ...