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A008846
Hypotenuses of primitive Pythagorean triangles.
55
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).
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
(Python) # for an array from the beginning
from math import gcd, isqrt
hypothenuses_upto = 433
A008846 = set()
for x in range(2, isqrt(hypothenuses_upto)+1):
for y in range(min(x-1, (yy:=isqrt(hypothenuses_upto-x**2))-(yy%2 == x%2)) , 0, -2):
if gcd(x, y) == 1: A008846.add(x**2 + y**2)
print(A008846:=sorted(A008846)) # Karl-Heinz Hofmann, Sep 30 2024
(Python) # for single k
from sympy import factorint
def A008846_isok(k): return not any([(pf-1) % 4 for pf in factorint(k)]) # Karl-Heinz Hofmann, Oct 01 2024
CROSSREFS
Subsequence of A004431 and of A000404 and of A339952; primitive elements: A002144.
Cf. A137409 (complement), disjoint union of A024409 and A120960.
Sequence in context: A020882 A081804 A004613 * A162597 A120960 A198440
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