

A008846


Hypotenuses of primitive Pythagorean triangles.


46



5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 425, 433
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OFFSET

1,1


COMMENTS

n for which there is no solution to 4/n = 2/x + 1/y for integers y > x > 0. Related to A073101.  T. D. Noe, Sep 30 2002
Discovered by Frenicle (on Pythagorean triangles): Methode pour trouver ..., page 14 on 44. First text of Divers ouvrages ... Par Messieurs de l'Academie Royale des Sciences, infolio, 6+518+1 pp., Paris, 1693. Also A020882 with only one of doubled terms (first: 65).  Paul Curtz, Sep 03 2008
All divisors of terms are of the form 4*k+1 (products of members of A002144).  Zak Seidov, Apr 13 2011
A024362(a(n)) > 0.  Reinhard Zumkeller, Dec 02 2012
Closed under multiplication. Primitive elements are in A002144.  JeanChristophe Hervé, Nov 10 2013
Not only the square of these numbers is equal to the sum of two nonzero squares, but the numbers themselves also are; this sequence is then a subsequence of A004431.  JeanChristophe Hervé, Nov 10 2013
All primitive Pythagorean triangles with sides {a, b, c} where a = a(n) are hypotenuses, "b" is the even leg, and "c" is the odd leg, have the property that sqrt(a+b) = t, which is an odd integer, and t divides c. This property derives from the fact that all primitive Pythagorean triangles can be from generated from two integers, j < i, as {j^2  i^2, 2*i*j, j^2 + i^2}. Also note: t = c only when the longer leg equals a(n)1.  Richard R. Forberg, May 11 2016


REFERENCES

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 10, 107.


LINKS

Zak Seidov, Table of n, a(n) for n = 1..87881 (with a(n) up to 10^6).
Ron Knott, Pythagorean Triples and Online Calculators


FORMULA

x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.


MAPLE

for x from 1 by 2 to 50 do for y from 2 by 2 to 50 do if gcd(x, y) = 1 then print(x^2+y^2); fi; od; od; [ then sort ].


MATHEMATICA

Union[ Map[ Plus@@(#1^2)&, Select[ Flatten[ Array[ {2*#1, 2*#21}&, {10, 10} ], 1 ], GCD@@#1 == 1& ] ] ] (* Olivier Gérard, Aug 15 1997 *)
lst = {}; Do[ If[ GCD[m, n] == 1, a = 2 m*n; b = m^2  n^2; c = m^2 + n^2; AppendTo[lst, c]], {m, 100}, {n, If[ OddQ@m, 2, 1], m  1, 2}]; Take[ Union@ lst, 57] (* Robert G. Wilson v, May 02 2009 *)
Union[Sqrt[#[[1]]^2+#[[2]]^2]&/@Union[Sort/@({Times@@#, (Last[#]^2First[#]^2)/2}&/@ (Select[Subsets[Range[1, 33, 2], {2}], GCD@@#==1&]))]] (* Harvey P. Dale, Aug 26 2012 *)


PROG

(Haskell)
a008846 n = a008846_list !! (n1)
a008846_list = filter f [1..] where
f n = all ((== 1) . (`mod` 4)) $ filter ((== 0) . (n `mod`)) [1..n]
 Reinhard Zumkeller, Apr 27 2011
(PARI) is(n)=Set(factor(n)[, 1]%4)==[1] \\ Charles R Greathouse IV, Nov 06 2015


CROSSREFS

Subsequence of A004431 and of A000404; primitive elements: A002144.
Cf. A020882, A073101.
Cf. A137409 (complement), disjoint union of A024409 and A120960.
Sequence in context: A020882 A081804 A004613 * A162597 A120960 A198440
Adjacent sequences: A008843 A008844 A008845 * A008847 A008848 A008849


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane, Ralph Peterson (RALPHP(AT)LIBRARY.nrl.navy.mil)


EXTENSIONS

More terms from T. D. Noe, Sep 30 2002


STATUS

approved



