A066326 a(1) = 5; for n > 1, a(n) is the least k > 0 not already included such that a(m)^2 + k^2 is a square for some m < n.

5, 12, 9, 16, 30, 35, 40, 42, 56, 33, 44, 63, 60, 11, 25, 32, 24, 7, 10, 18, 45, 28, 21, 20, 15, 8, 6, 36, 27, 48, 14, 55, 64, 70, 72, 54, 65, 75, 77, 80, 39, 52, 84, 13, 90, 91, 96, 99, 100, 105, 88, 66, 108, 81, 110, 112, 117, 120, 22, 50, 119, 126, 128, 132, 85, 135, 140

Offset

1,1

Comments

For a(1) = 0,1,2 no possible value of a(2) exists; for a(1) = 3,4 we get 4, 3 for a(2) but no further possible values. For a(1) >= 5 do we always get an infinite sequence?

From Robert Israel, Nov 22 2017: (Start)

Yes: if t >= 5 is the largest of a(1),...,a(n), then (if no smaller k works) it is always possible to take a(n+1) = (t^2-1)/2 if t is odd, t^2/4 - 1 if t is even.

Does the sequence include every number >= 5? (End)

Links

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

Example

a(5) = 30 because a(5)^2 + a(4)^2 = 30^2 + 16^2 = 34^2; a(6) = 35 because a(6)^2 + a(2)^2 = 35^2 + 12^2 = 37^2.

Maple

Cands:= {5}: S:= {}:
for n from 1 to 100 do
  A[n]:= min(Cands);
  Cands:= Cands minus {A[n]};
  if A[n]::odd then divs:= select(`<`, numtheory:-divisors(A[n]^2), A[n])
  else divs:= select(t -> t < A[n] and t::even and (A[n]^2/t)::even, numtheory:-divisors(A[n]^2))
  fi;
  Cands := Cands union (map(t -> (A[n]^2/t - t)/2, divs) minus S);
  S:= S union {A[n]};
od:
seq(A[i], i=1..100); # Robert Israel, Nov 22 2017

Crossrefs

Sequence in context: A009842 A169729 A070368 * A015242 A009415 A251935

Adjacent sequences: A066323 A066324 A066325 * A066327 A066328 A066329

Keyword

nonn,look

Author

Jonathan Ayres (jonathan.ayres(AT)ntlworld.com), Dec 15 2001

Extensions

New name from Robert Israel, Nov 22 2017

Status

approved