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"Binary prime squares": perfect squares written in base 2 which, considered whose binary expansions, read as decimal numbers, expansions, are primes.
Another definition: numbers having only digits 1 and 0, which, read in base 10 , are primes , and in base 2 are perfect squares. Derived from A108322 from which the 0-terms have been removed.
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fd2[n_]:=FromDigits[IntegerDigits[n, 2]]; fd2/@Select[Range[100300]^2, PrimeQ[fd2[#]]&](* Harvey P. Dale, Jul 10 2012 *)
fd2[n_]:=FromDigits[IntegerDigits[#, n, 2]]&; fd2/@Select[Range[300100]^2, PrimeQ[ FromDigits[ IntegerDigitsfd2[#, 2]]]&] (* Harvey P. Dale, Jul 10 2012 *)
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FromDigits[IntegerDigits[#, 2]]&/@Select[Range[100300]^2, PrimeQ[ FromDigits[ IntegerDigits[#, 2]]]&] (* Harvey P. Dale, Jul 10 2012 *)