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

1,1

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

Links

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

Example

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

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

Prog

(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)

Crossrefs

Cf. A016754.

Sequence in context: A154631 A199593 A356842 * A141552 A317720 A162822

Adjacent sequences: A174522 A174523 A174524 * A174526 A174527 A174528

Keyword

base,easy,nonn

Author

Michael R Peake, Mar 21 2010

Extensions

MATLAB program corrected by Robert Israel, Mar 14 2016

Status

approved