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

100, 144, 169, 200, 288, 300, 400, 441, 500, 528, 600, 700, 768, 800, 825, 867, 882, 900, 961, 1089, 1100, 1584, 2178, 2200, 3300, 4400, 4851, 5500, 6600, 7700, 8712, 8800, 9801, 9900, 10000, 10100, 10404, 10609, 10989

Offset: 1

Author

Reinhard Zumkeller, May 15 2002

Keywords

• nonn
• nice
• base

Comments

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

From Bernard Schott, Jan 02-10 2019: (Start)

There are six different families of integers in this sequence.

1) If k and rev(k) do not have the same number of digits:

All these integers are in A322835 where the first four families are explained and detailed.

Family 1: A002113(j) * 100^k

Family 2: A035090(j) * 100^k

Family 3: A082994(j) * 100^k

Family 4: A323061(j) * 10^(2k+1)

2) If k and rev(k) have the same number of digits.

All these integers are in A062917.

Family 5: Non-palindromic squares whose reverse is also square. These integers are in A035090.

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.

3) Relationships between these different sequences.

A035090 Union A082994 = A062917 with empty intersection, and,

A062917 Union A322835 = {This sequence} with empty intersection. (End)

Examples

			a(2)=144: rev(144)=441, 144*441=(12^2)*(21^2)=(12*21)^2 and 144<>12*21=252.
From _Bernard Schott_, Jan 02 2019: (Start)
Example for family 1: 200 * 2 = 400 = 20^2
Example for family 2: 14400 * 441 = 120^2 * 21^2 = 2520^2
Example for family 3: 28800 * 882 = (2 * 120^2) * (2 * 21^2) = 5040^2
Example for family 4: 5449680 * 869445 = 2176740^2
Example for family 5: 169 * 961 = 13^2 * 31^2 = 403^2
Example for family 6: 528 * 825 = (33 * 4^2) * (33 * 5^2) = 660^2. (End)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from Reinhard Zumkeller)

Crossrefs

Cf. A002113, A061205, A010052, A035090, A062917, A082994, A322835, A323061.

Programs

Haskell

a070760 n = a070760_list !! (n-1)
a070760_list = [x | x <- [0..], let y = a061205 x,
                    y /= x ^ 2, a010052 y == 1]
-- Reinhard Zumkeller, Apr 10 2012, Apr 29 2011

Mathematica

Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* Jean-François Alcover, Nov 30 2011 *)
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 *)