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A016070 Numbers k such that k^2 contains exactly 2 different digits, excluding 10^m, 2*10^m, 3*10^m. 5
4, 5, 6, 7, 8, 9, 11, 12, 15, 21, 22, 26, 38, 88, 109, 173, 212, 235, 264, 3114, 81619 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
FORMULA
A043537(a(n)) = 2. [Reinhard Zumkeller, Aug 05 2010]
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 *)
PROG
(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
CROSSREFS
Sequence in context: A173888 A341051 A120181 * A299536 A321025 A047569
KEYWORD
nonn,nice,base,more,hard
AUTHOR
STATUS
approved

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Last modified March 28 22:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)