Search: id:A009003
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%I A009003
%S A009003 5,10,13,15,17,20,25,26,29,30,34,35,37,39,40,41,45,50,51,52,53,55,58,60,
%T A009003 61,65,68,70,73,74,75,78,80,82,85,87,89,90,91,95,97,100,101,102,104,105,
%U A009003 106,109,110,111,113,115,116,117,119,120,122,123,125,130,135,136,137,140
%N A009003 Hypotenuse numbers (squares are sums of 2 distinct nonzero squares).
%C A009003 Multiples of Pythagorean primes A002144 or of primitive Pythagorean triangles' 
               hypotenuses A008846. - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 
               12 2003
%C A009003 It appears that this is exactly the sequence of positive integers with 
               at least one prime divisor of the form 4k+1. (This has been verified 
               for all terms<=500.) Compare A072592. - John W. Layman (layman(AT)math.vt.edu), 
               Mar 12 2008
%C A009003 The conjecture by Layman is correct. It is well known that the hypotenuses 
               of primitive Pythagorean triples are precisely those numbers with 
               all prime divisors of the form 4k+1. [From Franklin T. Adams-Watters 
               (FrankTAW(AT)Netscape.net), Apr 26 2009]
%D A009003 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
%H A009003 T. D. Noe, <a href="b009003.txt">Table of n, a(n) for n=1..1000</a>
%H A009003 R. Chapman, <a href="http://www.maths.ex.ac.uk/~rjc/courses/nt03/sumsquares.pdf">
               Pythagorean triples and sums of squares</a>
%H A009003 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/lr/lr.html">
               Landau-Ramanujan Constant</a>
%H A009003 Ron Knott, <a href="http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/
               pythag.html">Pythagorean Triples and Online Calculators</a>
%H A009003 <a href="Sindx_Su.html#ssq">Index entries for sequences related to sums 
               of squares</a>
%t A009003 f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++;
               If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[If[f[n^2]>0,AppendTo[lst,
               n]],{n,3,5!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Jun 15 2009]
%Y A009003 Cf. A009000, A009003, A024507, A004431. Complement of A004144.
%Y A009003 Primitive elements give A002144.
%Y A009003 Cf. A072592.
%Y A009003 Cf. A004613 [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), 
               Apr 26 2009]
%Y A009003 Sequence in context: A049197 A009000 A057100 this_sequence A071821 A084645 
               A092604
%Y A009003 Adjacent sequences: A009000 A009001 A009002 this_sequence A009004 A009005 
               A009006
%K A009003 nonn
%O A009003 1,1
%A A009003 David W. Wilson (davidwwilson(AT)comcast.net)

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