

A008846


Hypotenuses of primitive Pythagorean triangles.


50



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

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y)=1. Essentially the same as A004613.
Numbers 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 Frénicle (on Pythagorean triangles): Méthode pour trouver ..., page 14 on 44. First text of Divers ouvrages ... Par Messieurs de l'Académie 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
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
Conjecture: numbers p for which sqrt(1) exists in the padic numbering system. For example the 5adic number ...2431212, when squared, gives ...4444444, which is 1, and 5 is in the sequence.  Thierry Banel, Aug 19 2022


REFERENCES

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


LINKS



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]


CROSSREFS



KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS



STATUS

approved



