A024362 Number of primitive Pythagorean triangles with hypotenuse n.

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times C takes value n.

a(A137409(n)) = 0; a(A008846(n)) > 0; a(A120960(n)) = 1; a(A024409(n)) > 1; a(A159781(n)) = 4. - Reinhard Zumkeller, Dec 02 2012

If the formula given below is used one is sure to find all a(n) values for hypotenuses n <= N if the summation indices r and s are cut off at rmax(N) = floor((sqrt(N-4)+1)/2) and smax(N) = floor(sqrt(N-1)/2). a(n) is the number of primitive Pythagorean triples with hypotenuse n modulo catheti exchange. - Wolfdieter Lang, Jan 10 2016

References

• A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A020882, A024361, A046079, A046080.

Cf. A000290, A010052.

Cf. A001221, A002144, A106594.

Programs

Haskell

a024362 n = sum [a010052 y | x <- takeWhile (< nn) $ tail a000290_list,
                             let y = nn - x, y <= x, gcd x y == 1]
            where nn = n ^ 2
-- Reinhard Zumkeller, Dec 02 2012

Maple

f:= proc(n) local F;
   F:= numtheory:-factorset(n);
   if map(t -> t mod 4, F) <> {1} then return 0 fi;
   2^(nops(F)-1)
end proc:
seq(f(n),n=1..100); # Robert Israel, Jan 11 2016

Mathematica

Table[a0=IntegerExponent[n,2]; If[n==1 || a0>0, cnt=0, m=n/2^a0; p=Transpose[FactorInteger[m]][[1]]; c=Count[p, _?(Mod[#,4]==1 &)]; If[c==Length[p], cnt=2^(c-1), 0]]; cnt, {n,100}]
a[n_] := If[n==1||EvenQ[n]||Length[Select[FactorInteger[n], Mod[#[[1]], 4]==3 &]] >0, 0, 2^(Length[FactorInteger[n]]-1)]; Array[a, 100] (* Frank M Jackson, Jan 28 2018 *)

PARI

a(n)={my(m=0,k=n,n2=n*n,k2,l2);
while(1,k=k-1;k2=k*k;l2=n2-k2;if(l2>k2,break);if(issquare(l2),if(gcd(n,k)==1,m++)));  return(m);} \\ Stanislav Sykora, Mar 23 2015

Formula

a(n) = [q^n] T(q), n >= 1, where T(q) = Sum_{r>=1,s>=1} rpr(2*r-1, 2*s)*q^c(r,s), with rpr(k,l) = 1 if gcd(k,l) = 1, otherwise 0, and c(r,s) = (2*r-1)^2 + (2s)^2. - Wolfdieter Lang, Jan 10 2016

If all prime factors of n are in A002144 then a(n) = 2^(A001221(n)-1), otherwise a(n) = 0. - Robert Israel, Jan 11 2016

a(4*n+1) = A106594(n), other terms are 0. - Andrey Zabolotskiy, Jan 21 2022