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A016070 Numbers n such that n^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

Table of n, a(n) for n=1..21.

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-Fran├žois 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

Cf. A016069, A018884, A018885.

Sequence in context: A173888 A341051 A120181 * A299536 A321025 A047569

Adjacent sequences:  A016067 A016068 A016069 * A016071 A016072 A016073

KEYWORD

nonn,nice,base,more,hard

AUTHOR

Robert G. Wilson v

STATUS

approved

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Last modified April 11 18:59 EDT 2021. Contains 342888 sequences. (Running on oeis4.)