A272672 Numbers n such that the decimal concatenations 1n and 2n are both squares.

1025, 102500, 1390625, 10250000, 96700625, 139062500, 1025000000, 9670062500, 13906250000, 102500000000, 967006250000, 1390625000000, 10250000000000, 17654697265625, 96700625000000, 139062500000000, 910400191015625

Offset: 1

Author

Nathan Fox, Brooke Logan, and N. J. A. Sloane, May 20 2016

Keywords

• nonn
• base

Comments

The sequence is infinite because all the numbers 1025*100^k are members.

It would be nice to have the subsequence of "primitive" terms, those that do not end in an even number of zeros.

Let v be a number such that v^2 starts with 1. Let w^2 have the same digits as v^2 except that the initial digit is a 2. Then (w + v) * (w - v) = w^2 - v^2 = 10^m for some integer m. For the "primitive" terms, w + v turns out to be 250, 8000, 31250 etc. w - v turns out to be 40, 1250, 3200 etc. Given such w + v and w - v it is easy to find primitive elements. Furthermore, v must lie in (sqrt(11), sqrt(20)) * sqrt(10)^i and w must lie in (sqrt(21), sqrt(30)) * sqrt(10)^i for some integer i. - David A. Corneth, May 20 2016

Examples

			1025 is a member because 11025 = 105^2 and 21025 = 145^2.
		

Links

• Nathan Fox, Table of n, a(n) for n = 1..728
• Chai Wah Wu, Primitive terms < 10^1000

Crossrefs

Cf. A265432, A272671.

Programs

Magma

[n: n in [1..10000000 ] | IsSquare(Seqint(Intseq(n) cat Intseq(1))) and IsSquare(Seqint(Intseq(n) cat Intseq(2)))]; // Marius A. Burtea, Mar 21 2019

Maple

t1:=[];
for k from 1 to 2000000 do
if issqr(k+10^length(k)) and
issqr(k+2*10^length(k)) then t1:=[op(t1),k]; fi;
od;
t1;

PARI

is(n)=issquare(eval(Str(1,n))) && issquare(eval(Str(2,n))) \\ Charles R Greathouse IV, May 20 2016

Extensions

a(5)-a(17) from Alois P. Heinz, May 20 2016