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Numbers n such that all the following properties hold: (i) n*reverse(n) is a square; (ii) n != reverse(n); (iii) n and reverse(n) are not both squares; and (iv) n and reverse(n) have the same number of digits.
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%I #37 Jan 06 2019 13:25:58

%S 288,528,768,825,867,882,1584,2178,4851,8712,10989,13104,14544,15984,

%T 20808,21978,26208,27648,27848,36828,40131,44541,48139,48951,49686,

%U 57399,68694,80262,80802,82863,84672,84872,87912,93184,98901,99375

%N Numbers n such that all the following properties hold: (i) n*reverse(n) is a square; (ii) n != reverse(n); (iii) n and reverse(n) are not both squares; and (iv) n and reverse(n) have the same number of digits.

%C These numbers are counterexamples to the following conjecture given in the Ogilvy-Anderson reference: "When an integer and its reversal are unequal, their product is never a square except when both are squares." This sequence excludes terms like 2200, i.e. 2200*22 = 48400.

%C Contains x*(10^k+1) for k >= 3 with x in {144, 169, 288, 441, 528, 768, 825, 867, 882, 961}. - _Robert Israel_, Jun 11 2018

%C A035090 U {this sequence} = A062917, with empty intersection. - _Bernard Schott_, Jan 04 2019

%D C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.

%D J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 82-83. ASIN: B002ACVZ6O [From _Jason Earls_, Nov 22 2009]

%H Robert Israel, <a href="/A082994/b082994.txt">Table of n, a(n) for n = 1..168</a>

%H M. A. Rashid, M. A. Uppal, D. C. B. Marsh and A. Wayne, <a href="http://www.jstor.org/stable/2310179">Product of a Number and Its Reverse</a>, American Mathematical Monthly, vol. 64 (1957), p. 434, E-1243. - _Felix Fröhlich_, Jul 11 2014

%e a(5) = 867 because 867 * 768 = 665856 = 816^2.

%p revdigs:= proc(n) local L;

%p L:= convert(n,base,10);

%p add(L[-i]*10^(i-1),i=1..nops(L))

%p end proc:

%p filter:= proc(n) local r;

%p if issqr(n) then return false fi;

%p r:= revdigs(n);

%p r <> n and issqr(r*n) and not issqr(r);

%p end proc:

%p select(filter, [seq(seq(10*i+j,j=1..9),i=1..10^4)]); # _Robert Israel_, Jun 11 2018

%t Select[Range[10^5], And[UnsameQ @@ {#1, #2}, IntegerQ@ Sqrt[#1 #2], AllTrue[{#1, #2}, ! IntegerQ@ Sqrt@ # &], SameQ @@ (IntegerLength@ {#1, #2})] & @@ {#, IntegerReverse@ #} &] (* _Michael De Vlieger_, Jan 04 2019 *)

%Y Cf. A002113, A004086, A035090, A062917, A070760, A322835.

%K base,nonn

%O 1,1

%A _Jason Earls_, May 29 2003

%E Name clarified by _Bernard Schott_, Jan 04 2019