login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008846 Hypotenuses of primitive Pythagorean triangles. 40
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 (list; graph; refs; listen; history; text; internal format)
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,in-folio,6+518+1pp,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. - Jean-Christophe 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. - Jean-Christophe 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 is a divisible into 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^2,2ij,j^2+i^2}. Also note: t = c only when the longer leg equals to 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*#2-1}&, {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[#]^2-First[#]^2)/2}&/@ (Select[Subsets[Range[1, 33, 2], {2}], GCD@@#==1&]))]] (* Harvey P. Dale, Aug 26 2012 *)

PROG

(Haskell)

a008846 n = a008846_list !! (n-1)

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), 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified September 26 01:25 EDT 2017. Contains 292500 sequences.