A016069 Numbers k such that k^2 contains exactly 2 distinct digits.

4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 21, 22, 26, 30, 38, 88, 100, 109, 173, 200, 212, 235, 264, 300, 1000, 2000, 3000, 3114, 10000, 20000, 30000, 81619, 100000, 200000, 300000, 1000000, 2000000, 3000000, 10000000, 20000000

Offset

1,1

Comments

10^k, 2*10^k, 3*10^k for k > 0 are terms. - Chai Wah Wu, Dec 17 2021

Subsequence of primes is A057659. - Bernard Schott, Jul 29 2022

References

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

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..81

Eric Weisstein's World of Mathematics, Square Number

Formula

a(n) = ((n-1) mod 3 + 1)*10^(ceiling(n/3)-7) for n >= 34 (conjectured). - Chai Wah Wu, Dec 17 2021

Example

26 is in the sequence because 26^2 = 676 contains exactly 2 distinct digits.

Mathematica

Join[Select[Range[90000], Count[DigitCount[#^2], _?(#!=0&)]==2&], Flatten[ NestList[ 10#&, {100000, 200000, 300000}, 5]]] (* Harvey P. Dale, Mar 09 2013 *)
Select[Range[20000000], Length[Union[IntegerDigits[#^2]]]==2&] (* Vincenzo Librandi, Nov 04 2014 *)

Prog

(Haskell)
import Data.List (nub)
a016069 n = a016069_list !! (n-1)
a016069_list = filter ((== 2) . length . nub . show . (^ 2)) [0..]
-- Reinhard Zumkeller, Apr 14 2011
(PARI) /* needs version >= 2.6 */
for (n=1, 10^9, if ( #Set(digits(n^2))==2, print1(n, ", ") ) );
/* Joerg Arndt, Mar 09 2013 */
(Python)
from gmpy2 import is_square, isqrt
from itertools import combinations, product
A016069_list = []
for g in range(2, 10):
    n = 2**g-1
    for x in combinations('0123456789', 2):
        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):
                    A016069_list.append(int(isqrt(z)))
A016069_list = sorted(A016069_list) # Chai Wah Wu, Nov 03 2014
(Magma) [n: n in [0..20000000] | #Set(Intseq(n^2)) eq 2]; // Vincenzo Librandi, Nov 04 2014

Crossrefs

Cf. A016070, A018884, A018885, A057659.

Sequence in context: A191842 A299544 A039174 * A360210 A194283 A299546

Adjacent sequences: A016066 A016067 A016068 * A016070 A016071 A016072

Keyword

nonn,base,nice

Author

Robert G. Wilson v

Status

approved