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%I #120 Nov 04 2023 06:32:35
%S 5,13,17,25,29,37,41,53,61,65,65,73,85,85,89,97,101,109,113,125,137,
%T 145,145,149,157,169,173,181,185,185,193,197,205,205,221,221,229,233,
%U 241,257,265,265,269,277,281,289,293,305,305,313,317,325,325,337,349,353,365,365
%N Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles.
%C The largest member 'c' of the primitive Pythagorean triples (a,b,c) ordered by increasing c.
%C These are numbers of the form a^2 + b^2 where gcd(b-a, 2*a*b)=1. - _M. F. Hasler_, Apr 04 2010
%C Equivalently, numbers of the form a^2 + b^2 where gcd(a,b) = 1 and a and b are not both odd. To avoid double-counting, require a > b > 0. - _Franklin T. Adams-Watters_, Mar 15 2015
%C The density of such points in a circle with radius squared = a(n) is ~ Pi * a(n). Restricting to a > b > 0 reduces this by a factor of 1/8; requiring gcd(a,b)=1 provides a factor of 6/Pi^2; and a, b not both odd is a factor of 2/3. (2/3, not 3/4, because the case a, b both even has already been eliminated.) Multiplying, a(n) * Pi * 1/8 * 6/Pi^2 * 2/3 is a(n) / (2 * Pi). But n is approximately this number of points, so a(n) ~ 2 * Pi * n. Conjectured by _David W. Wilson_, proof by _Franklin T. Adams-Watters_, Mar 15 2015
%C Permutations are in A094194, A088511, A121727, A119321, A113482 and A081804. Entries of A024409 occur here more than once. - _R. J. Mathar_, Apr 12 2010
%C The distinct terms of this sequence seem to constitute a subset of the sequence defined as a(n) = (-1)^n + 6*n for n >= 1. - _Alexander R. Povolotsky_, Mar 15 2015
%C The terms in this sequence are given by f(m,n) = m^2 + n^2 where m and n are any two integers satisfying m > 1, n < m, the greatest common divisor of m and n is 1, and m and n are both not odd. E.g., f(m,n) = f(2,1) = 2^2 + 1^2 = 4 + 1 = 5. - _Agola Kisira Odero_, Apr 29 2016
%D M. de Frénicle, "Méthode pour trouver la solutions des problèmes par les exclusions", in: "Divers ouvrages de mathématiques et de physique, par Messieurs de l'Académie royale des sciences", Paris, 1693, pp 1-44.
%H David W. Wilson, <a href="/A020882/b020882.txt">Table of n, a(n) for n = 1..10000</a> (first 1593 terms from M. F. Hasler)
%H M. de Frénicle, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5493994j/f17.image">Méthode pour trouver la solutions des problèmes par les exclusions</a> (B.N.F. permanent link to a scan of the original edition).
%H Werner Hürlimann, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Huerlimann/huerli6.html">Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.
%H Hans Isdahl, <a href="https://web.archive.org/web/20160531122318/http://matematikk.nordreisavgs.net/fagstoff/pytagoras.htm">Pythagoras site (in Norwegian)</a>. [from Internet Archive Wayback Machine]
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>.
%H H. P. Robinson and N. J. A. Sloane, <a href="/A002037/a002037.pdf">Correspondence, 1971-1972</a>.
%H E. S. Rowland, <a href="https://ericrowland.github.io/investigations/tripleslist-long.html">Primitive Solutions to x^2 + y^2 = z^2</a>.
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rtritab.txt">Table of primitive Pythagorean triples and related parameters</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>.
%F a(n) = sqrt((A120681(n)^2 + A120682(n)^2)/2). - _Lekraj Beedassy_, Jun 24 2006
%F a(n) = sqrt(A046086(n)^2 + A046087(n)^2). - _Zak Seidov_, Apr 12 2011
%F a(n) ~ 2*Pi*n. - observation by _David W. Wilson_, proved by _Franklin T. Adams-Watters_ (cf. comments), Mar 15 2015
%F a(n) = sqrt(A180620(n)^2 + A231100(n)^2). - _Rui Lin_, Oct 09 2019
%t t={};Do[Do[a=Sqrt[c^2-b^2];If[a>b,Break[]];If[IntegerQ[a]&&GCD[a,b,c]==1,AppendTo[t,c]],{b,c-1,3,-1}],{c,400}];t (* _Vladimir Joseph Stephan Orlovsky_, Jan 21 2012 *)
%t f[c_] := Block[{a = 1, b, lst = {}}, While[b = Sqrt[c^2 - a^2]; a < b, If[ IntegerQ@ b && GCD[a, b, c] == 1, AppendTo[lst, a]]; a++]; lst]
%t Join @@ Table[ConstantArray[n, Length@f@n], {n, 1, 400, 4}] (* _Robert G. Wilson v_, Mar 16 2014; corrected by _Andrey Zabolotskiy_, Oct 31 2019 *)
%o (PARI) {my( c=0, new=[]); for( b=1,99, for( a=1, b-1, gcd(b-a,2*a*b) == 1 && new=concat(new,a^2+b^2)); new=vecsort(new); for( j=1,#new, new[j] > (b+1)^2 & (new=vecextract(new, Str(j,".."))) & next(2); write("b020882.txt",c++," "new[j])); new=[])} \\ _M. F. Hasler_, Apr 04 2010
%Y Cf. A004613, A008846, A020883-A020886, A046086, A046087, A222946 (as a number triangle).
%K nonn,easy
%O 1,1
%A _Clark Kimberling_
%E Edited by _N. J. A. Sloane_, May 15 2010
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