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

816326530612244897959183673469388, 1836734693877551020408163265306123, 3265306122448979591836734693877552, 3746097814776274713839750260145681581685744016649323621228

Offset: 1

Author

Giovanni Resta, Jan 25 2006

Keywords

• nonn
• base

Comments

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

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

Links

• Robert Israel, Table of n, a(n) for n = 1..113

Crossrefs

Cf. A102567, A106497, A115527 - A115556.

Programs

Maple

Res:= NULL:
for m from 1 to 67 do
if not numtheory:-issqrfree(3+10^m) then
   F:= select(t -> t[2]=1, ifactors(3+10^m)[2]);
   v:= mul(t[1], t=F);
   Res:= Res, seq(t^2*v, t = ceil(sqrt(10^(m-1)/(3*v))) .. floor(sqrt(10^m/(3*v))))
fi
od:
Res;  # Robert Israel, Aug 07 2019