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 A046081 Number of integer-sided right triangles with n as a hypotenuse or leg. 20
 0, 0, 1, 1, 2, 1, 1, 2, 2, 2, 1, 4, 2, 1, 5, 3, 2, 2, 1, 5, 4, 1, 1, 7, 4, 2, 3, 4, 2, 5, 1, 4, 4, 2, 5, 7, 2, 1, 5, 8, 2, 4, 1, 4, 8, 1, 1, 10, 2, 4, 5, 5, 2, 3, 5, 7, 4, 2, 1, 14, 2, 1, 7, 5, 8, 4, 1, 5, 4, 5, 1, 12, 2, 2, 9, 4, 4, 5, 1, 11, 4, 2, 1, 13, 8, 1, 5, 7, 2, 8, 5, 4, 4, 1, 5, 13, 2, 2, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Pythagorean triples including primitive ones and non-primitive ones. For a certain n, it may be a leg or the hypotenuse in either a primitive Pythagorean triple, or a non-primitive Pythagorean triple, or both. - Rui Lin, Nov 02 2019 REFERENCES A. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966. LINKS Lars Blomberg, Table of n, a(n) for n = 1..10000 Anonymous, Generator of all Pythagorean triples that include a given number [Internet Archive Wayback Machine] Ron Knott, Pythagorean Triples and Online Calculators F. Richman, Pythagorean Triples A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340. See Theorem 8. Eric Weisstein's World of Mathematics, Pythagorean Triple FORMULA a(n) = A046079(n) + A046080(n). - Lekraj Beedassy, Dec 01 2003 From Rui Lin, Nov 02 2019: (Start) a(n) = A024363(n) + A328949(n). a(n) = A024361(n) + A024362(n) + A328708(n) + A328712(n). (End) EXAMPLE From Rui Lin, Nov 02 2019: (Start) n=25 is the least number which meets all of following cases: 1. 25 is a leg of a primitive Pythagorean triple (25,312,313), so A024361(25)=1; 2. 25 is the hypotenuse of a primitive Pythagorean triple (7,24,25), so A024362(25)=1; 3. 25 is a leg of a non-primitive Pythagorean triple (25,60,65), so A328708(25)=1; 4. 25 is the hypotenuse of a non-primitive Pythagorean triple (15,20,25), so A328712(25)=1; 5. Combination 1. and 3. means A046079(25)=2; 6. Combination 2. and 4. means A046080(25)=2; 7. Combination 1. and 2. means A024363(25)=2; 8. Combination 3. and 4. means A328949(25)=2; 9. Combination of 1., 2., 3., and 4. means A046081(25)=4. (End) MATHEMATICA a[1] = 0; a[n_] := Module[{f}, f = Select[FactorInteger[n], Mod[#[[1]], 4] == 1&][[All, 2]]; (DivisorSigma[0, If[OddQ[n], n, n/2]^2]-1)/2 + (Times @@ (2*f+1) - 1)/2]; Array[a, 99] (* Jean-François Alcover, Jul 19 2017 *) PROG (PARI) a(n) = {oddn = n/(2^valuation(n, 2)); f = factor(oddn); for (k=1, #f~, if ((f[k, 1] % 4) != 1, f[k, 2] = 0); ); n1 = factorback(f); if (n % 2, (numdiv(n^2)+numdiv(n1^2))/2 -1, (numdiv((n/2)^2)+numdiv(n1^2))/2 -1); } \\ Michel Marcus, Mar 07 2016 CROSSREFS Cf. A006593, A046079, A046080, A024361, A024362, A024363, A328708, A328712, A328949. Sequence in context: A225332 A209032 A183935 * A190592 A062501 A182599 Adjacent sequences:  A046078 A046079 A046080 * A046082 A046083 A046084 KEYWORD nonn AUTHOR EXTENSIONS Improved name by Bernard Schott, Jan 03 2019 STATUS approved

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Last modified October 22 06:49 EDT 2020. Contains 337950 sequences. (Running on oeis4.)