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Numbers n such that (largest digit of n)^(smallest digit of n) + n is a square.
1

%I #34 Sep 14 2018 19:41:10

%S 0,13,72,80,91,120,137,163,188,251,275,281,317,321,360,388,391,440,

%T 495,527,627,840,891,960,1023,1088,1148,1151,1288,1363,1437,1520,1591,

%U 1674,1680,1757,1841,1927,2024,2113,2208,2303,2365,2400,2464,2491,2565,2600

%N Numbers n such that (largest digit of n)^(smallest digit of n) + n is a square.

%C Conjecture: 0 and 3844 are the only squares in this sequence.

%C There are only finitely many squares in the sequence because for some square big enough the offset to the next square is greater than 9^9. Now, for each number d = (largest digit of n)^(smallest digit of n) we just have to walk through the squares, check if this square has the correct (largest digit of n)^(smallest digit of n) property and added d is also a square. I have done that search computation exhaustively (using a PARI program). There are no more squares. - Maon Wenders, Jun 02 2012

%C This sequence has infinitely many terms because there are infinitely many numbers b^2 - 1 that contain a zero in their decimal expansion. - _T. D. Noe_, Jul 20 2012

%H Harvey P. Dale, <a href="/A096913/b096913.txt">Table of n, a(n) for n = 1..1000</a>

%e 388 is in the sequence because 8^3 + 388 = 900 = 30^2.

%t Join[{0},Select[Range[3000],IntegerQ[Sqrt[Max[IntegerDigits[#]]^ Min[ IntegerDigits[ #]]+ #]]&]] (* _Harvey P. Dale_, Sep 14 2018 *)

%Y Cf. A054054, A054055.

%K base,easy,nonn

%O 1,2

%A _Jason Earls_, Aug 17 2004