%I #98 Oct 02 2024 21:57:07

%S 5,13,17,25,29,37,41,53,61,65,73,85,89,97,101,109,113,125,137,145,149,

%T 157,169,173,181,185,193,197,205,221,229,233,241,257,265,269,277,281,

%U 289,293,305,313,317,325,337,349,353,365,373,377,389,397,401,409,421,425,433

%N Hypotenuses of primitive Pythagorean triangles.

%C 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.

%C 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

%C 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

%C All divisors of terms are of the form 4*k+1 (products of members of A002144). - _Zak Seidov_, Apr 13 2011

%C A024362(a(n)) > 0. - _Reinhard Zumkeller_, Dec 02 2012

%C Closed under multiplication. Primitive elements are in A002144. - _Jean-Christophe Hervé_, Nov 10 2013

%C 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

%C 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

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

%H Zak Seidov, <a href="/A008846/b008846.txt">Table of n, a(n) for n = 1..87881</a> (with a(n) up to 10^6).

%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>

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

%p 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 ].

%t Union[ Map[ Plus@@(#1^2)&, Select[ Flatten[ Array[ {2*#1, 2*#2-1}&, {10, 10} ], 1 ], GCD@@#1 == 1& ] ] ] (* _Olivier Gérard_, Aug 15 1997 *)

%t 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 *)

%t 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 *)

%o (Haskell)

%o a008846 n = a008846_list !! (n-1)

%o a008846_list = filter f [1..] where

%o f n = all ((== 1) . (`mod` 4)) $ filter ((== 0) . (n `mod`)) [1..n]

%o -- _Reinhard Zumkeller_, Apr 27 2011

%o (PARI) is(n)=Set(factor(n)[,1]%4)==[1] \\ _Charles R Greathouse IV_, Nov 06 2015

%o (Python) # for an array from the beginning

%o from math import gcd, isqrt

%o hypothenuses_upto = 433

%o A008846 = set()

%o for x in range(2, isqrt(hypothenuses_upto)+1):

%o for y in range(min(x-1, (yy:=isqrt(hypothenuses_upto-x**2))-(yy%2 == x%2)) , 0, -2):

%o if gcd(x,y) == 1: A008846.add(x**2 + y**2)

%o print(A008846:=sorted(A008846)) # _Karl-Heinz Hofmann_, Sep 30 2024

%o (Python) # for single k

%o from sympy import factorint

%o def A008846_isok(k): return not any([(pf-1) % 4 for pf in factorint(k)]) # _Karl-Heinz Hofmann_, Oct 01 2024

%Y Subsequence of A004431 and of A000404 and of A339952; primitive elements: A002144.

%Y Cf. A004613, A020882, A073101.

%Y Cf. A137409 (complement), disjoint union of A024409 and A120960.

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_, Ralph Peterson (RALPHP(AT)LIBRARY.nrl.navy.mil)

%E More terms from _T. D. Noe_, Sep 30 2002