A159781 -id:A159781 - cp's OEIS frontend

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

David W. Wilson

nonn

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

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

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

Cf. A020882, A024361, A046079, A046080.
Cf. A000290, A010052.
Cf. A001221, A002144, A106594.

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

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

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 *)

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

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

A024409 Hypotenuses of more than one primitive Pythagorean triangle.

Original entry on oeis.org

65, 85, 145, 185, 205, 221, 265, 305, 325, 365, 377, 425, 445, 481, 485, 493, 505, 533, 545, 565, 629, 685, 689, 697, 725, 745, 785, 793, 845, 865, 901, 905, 925, 949, 965, 985, 1025, 1037, 1073, 1105, 1145, 1157, 1165, 1189, 1205, 1241, 1261, 1285, 1313, 1325

Offset: 1

David W. Wilson

nonn

The subsequence allowing 4 different shapes is in A159781. [R. J. Mathar, Apr 12 2010]

A024362(a(n)) > 1. - Reinhard Zumkeller, Dec 02 2012

			65^2 = 16^2 + 63^2 = 33^2 + 56^2 (also = 25^2 + 60^2 = 39^2 + 52^2, but these are not primitive, with gcd = 5 resp. 13). Note that 65 = 1^2 + 8^2 = 4^2 + 7^2 is also the least integer > 1 to be a sum a^2 + b^2 with gcd(a,b) = 1 in two ways. - _M. F. Hasler_, May 18 2023

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A120960, subsequence of A008846.

import Data.List (findIndices)
a024409 n = a024409_list !! (n-1)
a024409_list = map (+ 1) $ findIndices (> 1) a024362_list
-- Reinhard Zumkeller, Dec 02 2012

f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[ Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range@ 335, f@# > 1 &] (* Robert G. Wilson v, Mar 16 2014 *)
Select[Tally[Sqrt[Total[#^2]]&/@Union[Sort/@({Times@@#,(Last[#]^2-First[ #]^2)/2}&/@(Select[Subsets[Range[1,71,2],{2}],GCD@@# == 1&]))]],#[[2]]> 1&][[All,1]]//Sort (* Harvey P. Dale, Sep 29 2018 *)

cp's OEIS Frontend

A024362 Number of primitive Pythagorean triangles with hypotenuse n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula