A115529 Numbers k such that the concatenation of 2*k with k gives a square.

1632653061224489796, 3673469387755102041, 6530612244897959184, 108166576527852893455922120064900, 130881557598702001081665765278529, 155759870200108166576527852893456, 182801514332071389940508382909681

Offset: 1

Author

Giovanni Resta, Jan 25 2006

Keywords

• nonn
• base

Comments

Numbers of the form k = a*b^2 where 10^(d-1) <= k < 10^d and (2*10^d+1)/a is a square. - Robert Israel, Jan 13 2021

Examples

			3265306122448979592_1632653061224489796 = 5714285714285714286^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..36

Crossrefs

Cf. A102567, A106497, A115527 - A115556.

Programs

Maple

f:= proc(d) local R,q,F,G,s,t,a,u,i;
   q:= 2*10^d+1;
   F:= ifactors(q)[2];
   G:= map(t -> [t[1],floor(t[2]/2)], F);
   s:= mul(t[1]^t[2],t=G);
   R:= NULL:
   for a in numtheory:-divisors(s) do
     u:= q/a^2;
     R:= R, seq(i^2*u, i=ceil(sqrt(10^(d-1)/u))..floor(sqrt((10^d-1)/u)))
   od;
   R
end proc:
seq(f(d),d=1..33); # Robert Israel, Jan 13 2021