A016070 Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m.

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

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• nice
• base
• more
• hard

Comments

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

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 109, p. 38, Ellipses, Paris 2008.
• R. K. Guy, Unsolved Problems in Number Theory, F24.

Links

• Michael Geißer, Theresa Körner, Sascha Kurz, and Anne Zahn, Squares with three digits, arXiv:2112.00444 [math.NT], 2021.
• Eric Weisstein's World of Mathematics, Square Number.

Crossrefs

Cf. A016069, A043537, A018884, A018885.

Programs

Mathematica

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

Python

from gmpy2 import is_square, isqrt
from itertools import combinations, product
A016070_list = []
for g in range(2,20):
    n = 2**g-1
    for x in combinations('0123456789',2):
        if not x in [('0','1'), ('0','4'), ('0','9')]:
            for i,y in enumerate(product(x,repeat=g)):
                if i > 0 and i < n and y[0] != '0':
                    z = int(''.join(y))
                    if is_square(z):
                        A016070_list.append(isqrt(z))
A016070_list = sorted(A016070_list) # Chai Wah Wu, Nov 03 2014

Formula

A043537(a(n)) = 2. [Reinhard Zumkeller, Aug 05 2010]