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 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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 FrÃ¶hlich, Jul 11 2014 EXAMPLE a(5) = 867 because 867 * 768 = 665856 = 816^2. 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 *) CROSSREFS Cf. A002113, A004086, A035090, A062917, A070760, A322835. Sequence in context: A280515 A061831 A250774 * A127350 A158253 A292054 Adjacent sequences:  A082991 A082992 A082993 * A082995 A082996 A082997 KEYWORD base,nonn AUTHOR Jason Earls, May 29 2003 EXTENSIONS Name clarified by Bernard Schott, Jan 04 2019 STATUS approved

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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)