cp's OEIS Frontend

A082994 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.

%I A082994 #37 Jan 06 2019 13:25:58
%S A082994 288,528,768,825,867,882,1584,2178,4851,8712,10989,13104,14544,15984,
%T A082994 20808,21978,26208,27648,27848,36828,40131,44541,48139,48951,49686,
%U A082994 57399,68694,80262,80802,82863,84672,84872,87912,93184,98901,99375
%N A082994 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 A082994 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 A082994 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 A082994 A035090 U {this sequence} = A062917, with empty intersection. - _Bernard Schott_, Jan 04 2019
%D A082994 C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
%D A082994 J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 82-83. ASIN: B002ACVZ6O [From _Jason Earls_, Nov 22 2009]
%H A082994 Robert Israel, <a href="/A082994/b082994.txt">Table of n, a(n) for n = 1..168</a>
%H A082994 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 A082994 a(5) = 867 because 867 * 768 = 665856 = 816^2.
%p A082994 revdigs:= proc(n) local L;
%p A082994   L:= convert(n,base,10);
%p A082994   add(L[-i]*10^(i-1),i=1..nops(L))
%p A082994 end proc:
%p A082994 filter:= proc(n) local r;
%p A082994   if issqr(n) then return false fi;
%p A082994   r:= revdigs(n);
%p A082994   r <> n and issqr(r*n) and not issqr(r);
%p A082994 end proc:
%p A082994 select(filter, [seq(seq(10*i+j,j=1..9),i=1..10^4)]); # _Robert Israel_, Jun 11 2018
%t A082994 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 A082994 Cf. A002113, A004086, A035090, A062917, A070760, A322835.
%K A082994 base,nonn
%O A082994 1,1
%A A082994 _Jason Earls_, May 29 2003
%E A082994 Name clarified by _Bernard Schott_, Jan 04 2019