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.

288, 528, 768, 825, 867, 882, 1584, 2178, 4851, 8712, 10989, 13104, 14544, 15984, 20808, 21978, 26208, 27648, 27848, 36828, 40131, 44541, 48139, 48951, 49686, 57399, 68694, 80262, 80802, 82863, 84672, 84872, 87912, 93184, 98901, 99375

Offset: 1

Author

Jason Earls, May 29 2003

Keywords

• base
• nonn

Comments

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.

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

A035090 U {this sequence} = A062917, with empty intersection. - Bernard Schott, Jan 04 2019

Examples

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

References

• C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY. (1966), pp. 88-89.
• J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 82-83. ASIN: B002ACVZ6O [From Jason Earls, Nov 22 2009]

Links

• Robert Israel, Table of n, a(n) for n = 1..168
• M. A. Rashid, M. A. Uppal, D. C. B. Marsh and A. Wayne, Product of a Number and Its Reverse, American Mathematical Monthly, vol. 64 (1957), p. 434, E-1243. - _Felix Fröhlich_, Jul 11 2014

Crossrefs

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

Programs

Maple

revdigs:= proc(n) local L;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local r;
  if issqr(n) then return false fi;
  r:= revdigs(n);
  r <> n and issqr(r*n) and not issqr(r);
end proc:
select(filter, [seq(seq(10*i+j,j=1..9),i=1..10^4)]); # Robert Israel, Jun 11 2018

Mathematica

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 *)

Extensions

Name clarified by Bernard Schott, Jan 04 2019