|
|
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
|
|
|
FORMULA
|
|
|
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
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))
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,base,more,hard
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|