A016070 - OEIS

%I #35 Dec 02 2021 09:25:42

%S 4,5,6,7,8,9,11,12,15,21,22,26,38,88,109,173,212,235,264,3114,81619

%N Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m.

%C No other terms below 3.16*10^20 (derived from A018884).

%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 109, p. 38, Ellipses, Paris 2008.

%D R. K. Guy, Unsolved Problems in Number Theory, F24.

%H Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, <a href="https://arxiv.org/abs/2112.00444">Squares with three digits</a>, arXiv:2112.00444 [math.NT], 2021.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.

%F A043537(a(n)) = 2. [_Reinhard Zumkeller_, Aug 05 2010]

%t Select[Range[100000],Length[DeleteCases[DigitCount[#^2],0]]==2 && !Divisible[ #,10]&] (* _Harvey P. Dale_, Aug 15 2013 *)

%t Reap[For[n = 4, n < 10^5, n++, id = IntegerDigits[n^2]; If[FreeQ[id, {_, 0 ...}], If[Length[Union[id]] == 2, Sow[n]]]]][[2, 1]] (* _Jean-François Alcover_, Sep 30 2016 *)

%o (Python)

%o from gmpy2 import is_square, isqrt

%o from itertools import combinations, product

%o A016070_list = []

%o for g in range(2,20):

%o ....n = 2**g-1

%o ....for x in combinations('0123456789',2):

%o ........if not x in [('0','1'), ('0','4'), ('0','9')]:

%o ............for i,y in enumerate(product(x,repeat=g)):

%o ................if i > 0 and i < n and y[0] != '0':

%o ....................z = int(''.join(y))

%o ....................if is_square(z):

%o ........................A016070_list.append(isqrt(z))

%o A016070_list = sorted(A016070_list) # _Chai Wah Wu_, Nov 03 2014

%Y Cf. A016069, A043537, A018884, A018885.

%K nonn,nice,base,more,hard

%O 1,1

%A _Robert G. Wilson v_