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A115531 Numbers k such that the concatenation of k with 3*k gives a square. 1


%S 816326530612244897959183673469388,1836734693877551020408163265306123,

%T 3265306122448979591836734693877552,

%U 3746097814776274713839750260145681581685744016649323621228

%N Numbers k such that the concatenation of k with 3*k gives a square.

%C If 3+10^m is not squarefree, say 3+10^m = u^2*v where v is squarefree, then the terms with length m are t^2*v where 10^m > 3*t^2*v >= 10^(m-1). The first m for which 3+10^m is not squarefree are 34, 59, 60, 61, 67. - _Robert Israel_, Aug 07 2019

%C Since 3+10^m is divisible by 7^2 for m = 34 + 42*k, the sequence contains 4*(3+10^m)/49, 9*(3+10^m)/49 and 16*(3+10^m)/49 for such m, and in particular is infinite. - _Robert Israel_, Aug 08 2019

%H Robert Israel, <a href="/A115531/b115531.txt">Table of n, a(n) for n = 1..113</a>

%p Res:= NULL:

%p for m from 1 to 67 do

%p if not numtheory:-issqrfree(3+10^m) then

%p F:= select(t -> t[2]=1, ifactors(3+10^m)[2]);

%p v:= mul(t[1], t=F);

%p Res:= Res, seq(t^2*v, t = ceil(sqrt(10^(m-1)/(3*v))) .. floor(sqrt(10^m/(3*v))))

%p fi

%p od:

%p Res; # _Robert Israel_, Aug 07 2019

%Y Cf. A102567, A106497, A115527 - A115556.

%K nonn,base

%O 1,1

%A _Giovanni Resta_, Jan 25 2006

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)