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A106594
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a(n) = number of primitive solutions to 4n+1 = x^2 + y^2.
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4
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1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 0, 1, 0, 1
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OFFSET
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1,16
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COMMENTS
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"Primitive" means that x and y are positive and mutually prime.
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LINKS
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EXAMPLE
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E.g. a(16)=2 because 65 = 8^2+1^2 = 7^2+4^2. a(11)=0 because although 45=6^2+3^2, 6 and 3 are not mutually prime. a(2)=0 because although 9=3^2+0^2, 0 is not positive.
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MAPLE
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local a, x, y, fourn;
fourn := 4*n+1 ;
a := 0 ;
for x from 1 do
if x^2 >= fourn then
return a;
else
y := fourn-x^2 ;
if issqr(y) then
y := sqrt(y) ;
if y <= x and igcd(x, y) = 1 then
a := a+1 ;
end if;
end if;
end if;
end do:
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MATHEMATICA
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Table[Length[If[CoprimeQ[#[[1]], #[[2]]], #, Nothing]&/@Union[Sort/@ ({#[[1, 2]], #[[2, 2]]}&/@FindInstance[{4 n+1==x^2+y^2, x>0, y>0}, {x, y}, Integers, 10])]], {n, 100}] (* Harvey P. Dale, Jun 29 2021 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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