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

1,1

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

Links

Nathan Fox, Table of n, a(n) for n = 1..728

Chai Wah Wu, Primitive terms < 10^1000

Example

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

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;

Prog

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

Crossrefs

Cf. A265432, A272671.

Sequence in context: A023002 A279643 A168119 * A180270 A103716 A291508

Adjacent sequences: A272669 A272670 A272671 * A272673 A272674 A272675

Keyword

nonn,base

Author

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

Extensions

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

Status

approved