A091569 a(1) = 1; for n > 1, a(n) is the smallest positive integer not already used such that a(n)*a(n-1) + 1 is a perfect square.

1, 3, 5, 7, 9, 11, 13, 15, 8, 6, 4, 2, 12, 10, 36, 34, 32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 52, 50, 48, 35, 33, 31, 29, 27, 25, 23, 21, 19, 17, 64, 62, 60, 40, 38, 148, 146, 144, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83

Offset

1,2

Comments

Does this sequence contain every positive integer? We could get an equally interesting sequence by choosing a(1) to be any other positive integer.

A sequence with the same condition but without the requirement for a(n) to be distinct would end up repeating (1,3) or (2,4), depending on the initial term. - Ivan Neretin, May 26 2015

Links

Ivan Neretin, Table of n, a(n) for n = 1..10000

Example

10 is followed by 36 because 10*36+1 = 19^2 and 8 and 12 were already used.

Maple

N:= 10^4: Used:= Vector(N, datatype=integer[4]):
a[1]:= 1: blocked:= false: Used[1]:= 1:
for n from 2 to 100 while not(blocked) do
    ndone:= false;
    if n = 2 then T:= [0]
      else T:= select(t -> t^2 mod a[n-1] = 1, [$0..a[n-1]-1])
      fi;
    for s from 0 while not (ndone) do
       for t in T while not (ndone) do
         x:= s * a[n-1] + t;
         if x <= 1 then next fi;
         y:= (x^2-1)/a[n-1];
         if y > N then blocked:= true; ndone:= true
         elif Used[y] = 0 then
            a[n]:= y;
            Used[y]:= 1;
            ndone:= true;
            print(n, y);
         fi
       od
    od
od:
seq(a[n], n=1..100); # Robert Israel, May 26 2015

Mathematica

a = {1}; Do[a = Join[a, Select[Complement[Range[(Max[a] + 1)*n], a], IntegerQ[Sqrt[#*a[[-1]] + 1]] &, 1]], {n, 2, 71}]; a (* Ivan Neretin, May 26 2015 *)

Prog

(MATLAB) A = zeros(1, 100); A(1) = 1; used = zeros(1, 1000); used(1) = 1; for i = 2:100; found = 0; k = 0; while found == 0; k = k + 1; if used(k) == 0; s = sqrt(k*A(i - 1) + 1); if s == floor(s); A(i) = k; used(k) = 1; found = 1; end; end; end; end; A

Crossrefs

Cf. A083203.

Sequence in context: A187472 A187411 A189401 * A206545 A293703 A120890

Adjacent sequences: A091566 A091567 A091568 * A091570 A091571 A091572

Keyword

easy,nonn

Author

David Wasserman, Mar 04 2004

Status

approved