A085305 - OEIS

%I #39 Sep 08 2022 08:45:11

%S 0,1,2,3,11,12,13,21,22,31,101,102,103,111,112,113,121,122,201,202,

%T 211,212,221,301,311,1001,1002,1003,1011,1012,1013,1021,1022,1031,

%U 1101,1102,1103,1111,1112,1113,1121,1122,1201,1202,1211,1212,1301,2001,2002,2011

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

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

%C Numbers (other than 0) that end in zero are excluded. - _N. J. A. Sloane_, Mar 20 2010

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

%H Reinhard Zumkeller, <a href="/A085305/b085305.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Sq#sqrev">Index entry for sequences related to reversing digits of squares</a>

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

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

%t 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}]

%t (* Second program: *)

%t 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 *)

%o a085305 n = a085305_list !! (n-1)

%o a085305_list = 0 : filter (\x -> x `mod` 10 > 0

%o                                  && a004086 (x^2) == (a004086 x)^2) [1..]

%o -- _Reinhard Zumkeller_, Jul 08 2011

%o (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

%o (PARI) isok(x) = (x==0) || ((x%10) && fromdigits(Vecrev(digits(x^2))) == fromdigits(Vecrev(digits(x)))^2); \\ _Michel Marcus_, Jan 11 2021

%Y Cf. A085306. See A061909 for another version.

%K base,nonn

%O 1,3

%A _Labos Elemer_, Jun 27 2003