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A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.

%I A070760 #76 Feb 24 2024 01:03:09
%S A070760 100,144,169,200,288,300,400,441,500,528,600,700,768,800,825,867,882,
%T A070760 900,961,1089,1100,1584,2178,2200,3300,4400,4851,5500,6600,7700,8712,
%U A070760 8800,9801,9900,10000,10100,10404,10609,10989
%N A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal.
%C A070760 If k is a palindrome (A002113), then 100*k is a term. If k is a term, then 100*k is a term. - _Chai Wah Wu_, Mar 31 2018
%C A070760 From _Bernard Schott_, Jan 02-10 2019: (Start)
%C A070760 There are six different families of integers in this sequence.
%C A070760 1) If k and rev(k) do not have the same number of digits:
%C A070760 All these integers are in A322835 where the first four families are explained and detailed.
%C A070760 Family 1: A002113(j) * 100^k
%C A070760 Family 2: A035090(j) * 100^k
%C A070760 Family 3: A082994(j) * 100^k
%C A070760 Family 4: A323061(j) * 10^(2k+1)
%C A070760 2) If k and rev(k) have the same number of digits.
%C A070760 All these integers are in A062917.
%C A070760 Family 5: Non-palindromic squares whose reverse is also square. These integers are in A035090.
%C A070760 Family 6: Non-palindromic numbers k, such that k * rev(k) is a square, with k and rev(k) not both square. These integers are in A082994.
%C A070760 3) Relationships between these different sequences.
%C A070760   A035090 Union A082994 = A062917 with empty intersection, and,
%C A070760 A062917 Union A322835 = {This sequence} with empty intersection. (End)
%H A070760 Chai Wah Wu, <a href="/A070760/b070760.txt">Table of n, a(n) for n = 1..10000</a> (n = 1..1000 from Reinhard Zumkeller)
%e A070760 a(2)=144: rev(144)=441, 144*441=(12^2)*(21^2)=(12*21)^2 and 144<>12*21=252.
%e A070760 From _Bernard Schott_, Jan 02 2019: (Start)
%e A070760 Example for family 1: 200 * 2 = 400 = 20^2
%e A070760 Example for family 2: 14400 * 441 = 120^2 * 21^2 = 2520^2
%e A070760 Example for family 3: 28800 * 882 = (2 * 120^2) * (2 * 21^2) = 5040^2
%e A070760 Example for family 4: 5449680 * 869445 = 2176740^2
%e A070760 Example for family 5: 169 * 961 = 13^2 * 31^2 = 403^2
%e A070760 Example for family 6: 528 * 825 = (33 * 4^2) * (33 * 5^2) = 660^2. (End)
%t A070760 Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* _Jean-François Alcover_, Nov 30 2011 *)
%t A070760 sdnQ[n_]:=Module[{c=n*IntegerReverse[n]},c!=n^2&&IntegerQ[Sqrt[c]]]; Select[ Range[11000],sdnQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 25 2016 *)
%o A070760 (Haskell)
%o A070760 a070760 n = a070760_list !! (n-1)
%o A070760 a070760_list = [x | x <- [0..], let y = a061205 x,
%o A070760                     y /= x ^ 2, a010052 y == 1]
%o A070760 -- _Reinhard Zumkeller_, Apr 10 2012, Apr 29 2011
%Y A070760 Cf. A002113, A061205, A010052, A035090, A062917, A082994, A322835, A323061.
%K A070760 nonn,nice,base
%O A070760 1,1
%A A070760 _Reinhard Zumkeller_, May 15 2002