

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.


1



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
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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^21)/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



