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Numbers k such that the digits of 4^k cannot be rearranged to form the digits of t^2, for t not a power of 2.
1

%I #18 Aug 25 2020 06:39:41

%S 0,1,2,3,8,9,11,12

%N Numbers k such that the digits of 4^k cannot be rearranged to form the digits of t^2, for t not a power of 2.

%C Leading zeros are not allowed.

%C 2^odd cannot be rearranged to a square number: odd powers of 2 are congruent to 2,5,8 mod 9; squares are congruent to 0,1,4,7 mod 9; and rearranging preserves the mod-9 value.

%C If it exists, a(9) > 78.

%D Don Reble, Posting to Sequence Fans Mailing List, Aug 21 2020

%e 4 is not here because 4^4 = 256 -> 625 = 25^2.

%e 10 is not here, because 4^10 = 1048576 -> 1056784 = 1028^2.

%e 11 is here, even though 4^11 = 4194304 -> 0413449 = 643^2, because leading zeros aren't allowed.

%Y Cf. A000302 (powers of 4), A337252.

%K nonn,base,more

%O 1,3

%A _N. J. A. Sloane_, Aug 22 2020