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%I #57 Jan 22 2023 15:01:20
%S 65,621770,281089082,2022652202,2042832002,868591084757,872546974178,
%T 872568754178,6979302951885,20313693904202,20313839704202,
%U 20331657922202,20331875722202,20333875702202,40313893704200
%N Rare numbers: k-r and k+r are both perfect squares, where r is reverse of k and k is non-palindromic.
%C There are 124 terms up to 10^22. - _Shyam Sunder Gupta_, Dec 15 2019
%C All non-palindromic numbers m = a^2 + b^2 such that reversal(m) = 2*a*b are terms of this sequence. For the numbers with this property, m - reversal(m) = (a-b)^2 and m + reversal(m) = (a+b)^2. - _Metin Sariyar_, Dec 19 2019
%C There are 132 terms up to 10^23. - _Shyam Sunder Gupta_, Nov 25 2021
%D Shyam Sunder Gupta, Systematic computations of rare numbers, The Mathematics Education, Vol. XXXII, No. 3, Sept. 1998.
%H Shyam Sunder Gupta, <a href="/A035519/b035519.txt">Table of n, a(n) for n = 1..132</a>
%H Shyam Sunder Gupta, <a href="http://www.shyamsundergupta.com/rare.htm">Rare Numbers</a>
%H R. K. Guy, <a href="http://www.jstor.org/stable/2325149">Conway's RATS and other reversals</a>, Unsolved Problems Column, American Math. Monthly, page 425, May 1989.
%H R. K. Guy, <a href="http://www.jstor.org/stable/2324584">Unsolved problems come of Age</a>, American Math. Monthly, page 908, Dec. 1989.
%H Carlos Rivera, <a href="https://www.primepuzzles.net/conjectures/conj_023.htm">Conjecture 23. The Shyam's conjecture about the Rare Numbers</a>, The Prime Puzzles and Problems Connection.
%e 65 - 56 = 9 and 65 + 56 = 121 are both squares.
%t r[n_]:=FromDigits[Reverse[IntegerDigits[n,10]],10]; f[n_]:=n!=r[n]&&IntegerQ[Sqrt[n-r[n]]]&&IntegerQ[Sqrt[n+r[n]]]; Timing[lst={};Do[If[f[n],AppendTo[lst,n]],{n,11,15!}];lst] (* _Vladimir Joseph Stephan Orlovsky_, Oct 10 2009 *)
%t Select[Range[2043*10^6],!PalindromeQ[#]&&AllTrue[{Sqrt[#+ IntegerReverse[ #]],Sqrt[ #-IntegerReverse[#]]},IntegerQ]&] (* The program generates the first 5 terms of the sequence. *) (* _Harvey P. Dale_, Jan 22 2023 *)
%o (PARI) isok(k) = {my(d = digits(k), rd = Vecrev(d), r = fromdigits(rd)); (d != Vecrev(d)) && issquare(k-r) && issquare(k+r);} \\ _Michel Marcus_, Jan 06 2020
%Y Cf. A059755.
%K nonn,base,nice
%O 1,1
%A _Shyam Sunder Gupta_, Dec 11 1999
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