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COMMENT
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Note that there are no such primes in any range ]2^(2n-1),2^2n], as all even-length binary palindromes are divisible by three (cf. A048702).
Ratio a(n)/A036378(2n) converges as follows: 1, 0.4, 0.230769, 0.069767, 0.051095, 0.025862, 0.014268, 0.007006, 0.00461, 0.00193, 0.00101, 0.000487, 0.000235, 0.000127, 0.000061, 0.000029
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