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%I #39 Jun 16 2020 16:27:45
%S 544968,547658,560106,601065,856745,869445,2495295,4601685,5606106,
%T 5861064,5925942,6016065,20861005,21778875,22972005,29389855,42251835,
%U 50016802,50027922,51826326,53815224,55898392,56066106,56570706,56873466,57887712,60166065,60707565
%N Numbers n that are not multiples of 10 and such that 10*n*R(n) is a square, where R(n) = A004086(n) is the decimal digits of n reversed.
%C 10*a(n) are exactly the terms in A322835 that are not multiples of 100.
%C m is a term if and only if R(m) is a term.
%C The product of the first and last digits of a term is a multiple of 10, i.e., the first and last digits of a term are the digit 5 and an even nonzero digit.
%C The sequence has an infinite number of terms. For instance, 601x065 is a term where x is a string of k repeated digits 6 and k >= 0, i.e., 601065, 6016065, 60166065, etc. Similarly numbers of the form 560x106 are also terms.
%C To see this, let a = 601*10^(3+k) + 65 + 6000*(10^k-1)/9. Then R(a) = 56*10^(4+k) + 106 + 6000*(10^k-1)/9. The number 10*a*R(a) can be written as 30360100*(10^(k + 3) - 1)^2/9 whose square root is 5510*(10^(k + 3) - 1)/3.
%C From _Chai Wah Wu_, Feb 18 2019: (Start)
%C 22994x77005 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 22994*10^(5+k) + 77005 + 100000*(10^k-1). Then R(a) = 50077*10^(5+k) + 49922 + 100000*(10^k-1). The number 10*a*R(a) can be written as 11515436100*(10^(k+5) - 1)^2, whose square root is 107310*(10^(k+5) - 1).
%C 23804x76195 is a term where x is a string of k repeated digits 9 and k >= 0. Let a = 23804*10^(5+k) + 76195 + 100000*(10^k-1). Then R(a) = 59167*10^(5+k) + 40832 + 100000*(10^k-1). The number 10*a*R(a) can be written as 14084942400*(10^(k+5) - 1)^2, whose square root is 118680*(10^(k+5) - 1).
%C If w is a term with decimal representation a, then the number n corresponding to the string axa is also a term, where x is a string of k repeated digits 0 and k >= 0. The number n = w*10^(k+m)+w = w*(10^(k+m)+1) where m is the number of digits of w. Note that n is also not a multiple of 10. Then R(n) = R(w)*10^(k+m)+R(w) = R(w)(10^(k+m)+1). Then 10*n*R(n) = 10*w*R(w)(10^(k+m)+1)^2 which is a square since w is a term.
%C The same argument shows that numbers corresponding to axaxa, axaxaxa, ..., are also terms.
%C For example, since 544968 is a term, so are 544968544968, 5449680544968, 54496800544968, 5449680054496800544968, etc.
%C (End)
%H Giovanni Resta, <a href="/A323061/b323061.txt">Table of n, a(n) for n = 1..200</a> (first 54 terms from Chai Wah Wu)
%e 238026195 * 591620832 * 10 = 1186681320^2.
%t Select[Range[61*10^6],Mod[#,10]!=0&&IntegerQ[Sqrt[10# IntegerReverse[ #]]]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jun 16 2020 *)
%o (PARI) isok(n) = (n % 10) && issquare(10*n*fromdigits(Vecrev(digits(n)))); \\ _Michel Marcus_, Jan 10 2019
%Y Cf. A004086, A322835.
%K nonn,base
%O 1,1
%A _Chai Wah Wu_, Jan 07 2019
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