A263609 Base-4 numbers whose square is a palindrome in base 4.
0, 1, 11, 101, 111, 1001, 1013, 1103, 10001, 10101, 10121, 10331, 100001, 100133, 1000001, 1001001, 1001201, 1010301, 1100211, 1100323, 1101211, 10000001, 10001333, 10013201, 10031113, 100000001, 100010001, 100012001, 100103001, 100301113, 100332101, 101002101, 103231203, 110002011
Offset: 1
Examples
From _Mattew Bondar_, Mar 12 2021: (Start) 111_4 = 21_10, 21^2 = 441, 441_10 = 12321_4 (palindrome). 1013_4 = 71_10, 71^2 = 5041, 5041_10 = 1032301_4 (palindrome). (End)
Links
- G. J. Simmons, On palindromic squares of non-palindromic numbers, J. Rec. Math., 5 (No. 1, 1972), 11-19. [Annotated scanned copy]
Programs
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Mathematica
FromDigits /@ IntegerDigits[Select[Range[0, 2^17], PalindromeQ@ IntegerDigits[#^2, 4] &], 4] (* Michael De Vlieger, Mar 13 2021 *)
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Python
def decimal_to_quaternary(n): if n == 0: return '0' b = '' while n > 0: b = str(n % 4) + b n = n // 4 return b x = 0 counter = 0 while True: y = decimal_to_quaternary(x ** 2) if y == y[::-1]: print(int(decimal_to_quaternary(x))) counter += 1 x += 1 # Mattew Bondar, Mar 10 2021