A085305 Numbers such that first reversing digits and then squaring equals the result of first squaring and then reversing.

0, 1, 2, 3, 11, 12, 13, 21, 22, 31, 101, 102, 103, 111, 112, 113, 121, 122, 201, 202, 211, 212, 221, 301, 311, 1001, 1002, 1003, 1011, 1012, 1013, 1021, 1022, 1031, 1101, 1102, 1103, 1111, 1112, 1113, 1121, 1122, 1201, 1202, 1211, 1212, 1301, 2001, 2002, 2011

Offset: 1

Author

Labos Elemer, Jun 27 2003

Keywords

• base
• nonn

Comments

Only digits {0, 1, 2, 3} seem to arise.

Numbers (other than 0) that end in zero are excluded. - N. J. A. Sloane, Mar 20 2010

Examples

			n = 13 is a term because 31^2 = 961 = rev(169) = rev(13^2) = rev(rev(31)^2).
		

References

• David Wells, The Dictionary of Curious and Interesting Numbers. London: Penguin Books (1997): p. 124.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Index entry for sequences related to reversing digits of squares

Crossrefs

Cf. A085306. See A061909 for another version.

Programs

Magma

[0] cat [ m: n in [1..1810] | Reverse(Intseq(m^2)) eq Intseq(Seqint(Reverse(Intseq(m)))^2) where m is n+Floor((n-1)/9) ];  // Bruno Berselli, Jul 08 2011

Mathematica

rt[x_] := tn[Reverse[IntegerDigits[x]]] Do[s = rt[n^2]; s1=rt[n]^2; If[Equal[s, s1]&&!Equal[Mod[n, 10], 0], Print[{n, s, rt[s1]}]], {n, 0, 1000000}]
(* Second program: *)
Select[Range[0, 1999], Mod[#,10] != 0 && FromDigits[Reverse[IntegerDigits[#^2]]] == FromDigits[Reverse[IntegerDigits[#]]]^2 &] (* Alonso del Arte, Oct 08 2012; corrected by Jean-François Alcover, Jan 11 2021 *)

PARI

isok(x) = (x==0) || ((x%10) && fromdigits(Vecrev(digits(x^2))) == fromdigits(Vecrev(digits(x)))^2); \\ Michel Marcus, Jan 11 2021

Formula

Solutions to rev(x^2) = rev(x)^2.