cp's OEIS Frontend

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

%I A272672 #40 Sep 08 2022 08:46:16
%S A272672 1025,102500,1390625,10250000,96700625,139062500,1025000000,
%T A272672 9670062500,13906250000,102500000000,967006250000,1390625000000,
%U A272672 10250000000000,17654697265625,96700625000000,139062500000000,910400191015625
%N A272672 Numbers n such that the decimal concatenations 1n and 2n are both squares.
%C A272672 The sequence is infinite because all the numbers 1025*100^k are members.
%C A272672 It would be nice to have the subsequence of "primitive" terms, those that do not end in an even number of zeros.
%C A272672 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
%H A272672 Nathan Fox, <a href="/A272672/b272672.txt">Table of n, a(n) for n = 1..728</a>
%H A272672 Chai Wah Wu, <a href="/A272672/a272672.txt">Primitive terms < 10^1000</a>
%e A272672 1025 is a member because 11025 = 105^2 and 21025 = 145^2.
%p A272672 t1:=[];
%p A272672 for k from 1 to 2000000 do
%p A272672 if issqr(k+10^length(k)) and
%p A272672 issqr(k+2*10^length(k)) then t1:=[op(t1),k]; fi;
%p A272672 od;
%p A272672 t1;
%o A272672 (PARI) is(n)=issquare(eval(Str(1,n))) && issquare(eval(Str(2,n))) \\ _Charles R Greathouse IV_, May 20 2016
%o A272672 (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
%Y A272672 Cf. A265432, A272671.
%K A272672 nonn,base
%O A272672 1,1
%A A272672 _Nathan Fox_, _Brooke Logan_, and _N. J. A. Sloane_, May 20 2016
%E A272672 a(5)-a(17) from _Alois P. Heinz_, May 20 2016