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

1,3

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

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

Formula

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

Example

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

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 *)

Prog

a085305 n = a085305_list !! (n-1)
a085305_list = 0 : filter (\x -> x `mod` 10 > 0
                                 && a004086 (x^2) == (a004086 x)^2) [1..]
-- Reinhard Zumkeller, Jul 08 2011
(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
(PARI) isok(x) = (x==0) || ((x%10) && fromdigits(Vecrev(digits(x^2))) == fromdigits(Vecrev(digits(x)))^2); \\ Michel Marcus, Jan 11 2021

Crossrefs

Cf. A085306. See A061909 for another version.

Sequence in context: A007932 A334054 A035122 * A189818 A116032 A116029

Adjacent sequences: A085302 A085303 A085304 * A085306 A085307 A085308

Keyword

base,nonn

Author

Labos Elemer, Jun 27 2003

Status

approved