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A070760 Numbers k such that k*rev(k) is a square different from k^2, where rev=A004086, decimal reversal. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)

LINKS

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

EXAMPLE

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)

MAPLE

T:=proc(n) local i, x: x:=convert(n, base, 10):

add(x[-i]*10^(i-1), i=1..nops(x)) end:

op(select(n->frac(sqrt(n*T(n)))=0 and n^2<>n*T(n), [$1..10989]));

# Paolo P. Lava, Feb 20 2019

MATHEMATICA

Select[ Range[11000], (k = Sqrt[ # * FromDigits @ Reverse @ IntegerDigits[#]]; IntegerQ[k] && k != #) &] (* Jean-Fran├žois 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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A108551 A319440 A096598 * A161902 A153673 A258482

Adjacent sequences:  A070757 A070758 A070759 * A070761 A070762 A070763

KEYWORD

nonn,nice,base

AUTHOR

Reinhard Zumkeller, May 15 2002

STATUS

approved

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Last modified October 23 13:32 EDT 2021. Contains 348214 sequences. (Running on oeis4.)