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(Python)
from sympy import isprime
def ok(n): return isprime(n) and (b:=bin(n)[2:]) == b[::-1]
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Apr 20 2024
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Number of terms less than 4^k, k=0,1,2,3,...: 2, 4, 6, 9, 12, 19, 31, 54, 94, 188, 330, 601, 1081, 1937, 3658, 6757, 12329, 23128, 43910, 83378, 156050, 295917, 570397, 1090773, 2077091, 3991188, 7717805, 14825248, 28507573, 54938370, 106350935, 1, 3, 5, 8, 11, 18, 30, 53, 93, 187, 329, 600, 1080, 1936, 3657, 6756, 12328, 23127, 43909, 83377, 156049, 295916, 570396, 1090772, 2077090, 3991187, 7717804, 14825247, 28507572, 54938369, 106350934, ..., partial sums of A095741 plus 21. - Robert G. Wilson v, Feb 23 2018, corrected by _Jeppe Stig Nielsen_, Jun 17 2023
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(MAGMAMagma) [NthPrime(n): n in [1..5000] | (Intseq(NthPrime(n), 2) eq Reverse(Intseq(NthPrime(n), 2)))]; // Vincenzo Librandi, Feb 24 2018
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