A008846 Hypotenuses of primitive Pythagorean triangles.

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

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, in-folio, 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. - 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

Conjecture: numbers p for which sqrt(-1) exists in the p-adic numbering system. For example the 5-adic 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

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 and of A339952; primitive elements: A002144.

Cf. A004613, 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