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Squares whose digital rotation is also a square.
1

%I #51 Apr 10 2024 19:08:41

%S 1,121,196,529,625,961,6889,10201,69169,1002001,1022121,1212201,

%T 5221225,100020001,100220121,109181601,121022001,522808225,602555209,

%U 10000200001,10002200121,10020210201,10201202001,12100220001,62188888129,1000002000001

%N Squares whose digital rotation is also a square.

%C From _Jon E. Schoenfield_, Nov 13 2016: (Start)

%C It is assumed that the rotation changes each digit 6 to a 9 and vice versa, and that the digits 0, 1, 2, 5, and 8 are unchanged by the rotation, as is the case with a seven-segment display in which the digits are formed basically as follows:

%C . _ _ _ _ _ _ _ _

%C | | | _| _| |_| |_ |_ | |_| |_|

%C |_| | |_ _| | _| |_| | |_| _|

%C (End)

%C Sequence is infinite since (10^m+1)^2 for m>0 are terms. Leading zeros after rotation are not allowed, as otherwise 10^m would be terms. All terms start and end with digits 1, 5, 6 or 9. - _Chai Wah Wu_, Apr 09 2024

%H Chai Wah Wu, <a href="/A275028/b275028.txt">Table of n, a(n) for n = 1..164</a>

%H Prime Curios, <a href="https://primes.utm.edu/curios/page.php/31.html">31</a>

%e 961 becomes 196 under such a rotation.

%t Select[Range[10^6]^2, If[Or[IntersectingQ[#, {3, 4, 7}], Last@# == 0], False, IntegerQ@ Sqrt@ FromDigits[Reverse@ # /. {6 -> 9, 9 -> 6}]] &@ IntegerDigits@ # &] (* _Michael De Vlieger_, Nov 14 2016 *)

%o (Python)

%o from itertools import count, islice

%o from sympy.ntheory.primetest import is_square

%o def A275028_gen(): # generator of terms

%o r, t = ''.maketrans('69','96'), set('0125689')

%o for l in count(1):

%o if l%10:

%o m = l**2

%o if set(s:=str(m)) <= t and is_square(int(s[::-1].translate(r))):

%o yield m

%o A275028_list = list(islice(A275028_gen(),10)) # _Chai Wah Wu_, Apr 09 2024

%Y Cf. A178316.

%K nonn,base

%O 1,2

%A _Seiichi Manyama_, Nov 12 2016