A284986 Integers k such that k^2 is not a palindrome and the square root of the reverse of k^2 yields an integer with the same number of digits as k that is not the reverse of k.

33, 99, 3168, 6501, 20508, 21468, 110922, 219111, 303577, 304877, 1100922, 1109211, 1110922, 1119111, 2191011, 2191111, 3080367, 3140793, 10110922, 11009111, 11009122, 11009221, 11019111, 11091011, 11091022, 11091111, 11091121, 11091122, 11091202, 11091211

Offset: 1

Author

Geoffrey Marnell, Apr 06 2017

Keywords

• nonn
• base

Comments

Computer-based iteration shows that there are very few integers such that the square root of the reverse of the square of the integer yields an integer with the same number of digits. Between 1 and 100,000 there are just 168. Curiously, 153 of these 168 yield an integer that is the mirror (or reverse) of the starting number. Two examples: 21^2 = 441 and sqrt(144) = 12; 22^2 = 484 and sqrt(484) = 22. Intuition suggests that this mirroring is more likely if the square is a palindrome (as in the second example). But this is not the case. (The square is not a palindrome in 132 of the 153 cases.) The opposite appears to be the true for those integers that do not yield an equi-digit mirror: more cases occur where the square is a palindrome (9) than when it is not (6). The sequence is of those integers that generate neither a palindrome on squaring nor a mirror when the square root is taken of the reverse of the square (up to 10^6).

References

• G. Marnell, Mathematical Doodlings: Curiosities, Conjectures and Challenges, Burdock Books, 2017, pages 96-102.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..9794 (terms < 3*10^13)
• Index entry for sequences related to reversing digits of squares

Crossrefs

Subsequence of A102859.

Programs

Mathematica

rev[n_] := FromDigits@ Reverse@ IntegerDigits@n; Select[Range[10^5], (r = rev[#^2]) != #^2 && IntegerQ[q = Sqrt[r]] && q != rev[#] && Equal @@ IntegerLength[{q, #}] &] (* Giovanni Resta, Apr 15 2017 *)

Extensions

a(11)-a(30) from Giovanni Resta, Apr 15 2017