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 A024362 Number of primitive Pythagorean triangles with hypotenuse n. 19
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,65 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 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 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 *) PROG (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 (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 CROSSREFS Cf. A020882, A024361, A046079, A046080. Cf. A000290, A010052. Cf. A001221, A002144, A106594. Sequence in context: A341755 A245515 A327170 * A347245 A104488 A244413 Adjacent sequences: A024359 A024360 A024361 * A024363 A024364 A024365 KEYWORD nonn AUTHOR David W. Wilson STATUS approved

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Last modified May 28 14:15 EDT 2023. Contains 363019 sequences. (Running on oeis4.)