A174525 Bases N in which ab and ba are different squares, for some a and b.

9, 12, 17, 19, 24, 25, 28, 33, 40, 49, 51, 52, 57, 60, 64, 67, 72, 73, 79, 81, 84, 88, 89, 96, 97, 99, 103, 105, 108, 112, 115, 116, 121, 124, 129, 134, 136, 144, 145, 148, 156, 161, 163, 168, 169, 172, 177, 180, 184, 192, 193, 199

Offset: 1

Author

Michael R Peake, Mar 21 2010

Keywords

• base
• easy
• nonn

Comments

From Robert Israel, Mar 14 2016: (Start)

Leading 0's are not allowed.

Conjecture: all odd squares (A016754) except 1 are terms of the sequence. (End)

N=(2n+1)^2, a=n^2, b=4n^2+2n+1 shows that (2n+1)^2 is a term, so this sequence is infinite. - Michael R Peake, Mar 21 2017

Examples

			17_9 and 71_9 are squares. 14_12 and 41_12 are squares.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A016754.

Programs

MATLAB

Match = zeros(1,100);
for N=2:200, Tens=zeros(1,N-1);Units=zeros(1,N-1); for a=N-1:-1:sqrt(N),c=a^2;Tens(a)=floor(c/N);Units(a)=rem(c,N);end; for a=N-1:-1:sqrt(N),h=find((Units==Tens(a))&([1:N-1]~=a)); if length(h),Match=any(Units(a)==Tens(h)); if Match,Sol(N)=Sol(N)+1;end;end;end;end;
find(Match > 0)

Maple

filter:= proc(n) local x,a,b,R;
    for x from ceil(sqrt(n)) to n-1 do
      a:= x^2 mod n;
      if a=0 then next fi;
      b:= (x^2-a)/n;
      if assigned(R[b,a]) then return true fi;
      R[a,b]:= 1;
    od;
    false
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 14 2016

Extensions

MATLAB program corrected by Robert Israel, Mar 14 2016