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%I #22 Feb 21 2023 02:09:31
%S 15,30,35,39,51,55,60,70,75,78,87,91,95,102,105,110,111,115,119,120,
%T 123,135,140,143,150,155,156,159,165,174,175,182,183,187,190,195,203,
%U 204,210,215,219,220,222,230,235,238,240,246,247,255,259,267,270,273,275
%N Hypotenuse numbers A009003 which cannot be represented as sum of 2 distinct nonzero squares.
%C Numbers with both at least one prime factor of form 4k+1 (which makes the square decomposable into the sum of two squares), and with at least one prime factor of form 4k+3 to an odd multiplicity (which makes the number itself not decomposable). This is a direct consequence of Fermat's Christmas theorem on the sum of two squares (Fermat announced its proof - without giving it - in a letter to Mersenne dated December 25, 1640). - _Jean-Christophe Hervé_, Nov 19 2013
%C Numbers n such that n^2 is the sum of two nonzero squares while n is not. Also note that sequence is equivalent to "Hypotenuse numbers A009003 which cannot be represented as sum of 2 nonzero squares." The reason is, if n is the sum of two nonzero squares in exactly one way and n = a^2 + a^2, then n^2 cannot be the sum of two nonzero squares. - _Altug Alkan_, Apr 14 2016
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquareNumber.html">Square Number</a>
%H Proof Wiki, <a href="http://www.proofwiki.org/wiki/Fermat%27s_Christmas_Theorem">Fermat's Christmas Theorem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares">Fermat's Theorem on sums of two squares</a>
%H Wikipedia (fr), <a href="http://fr.wikipedia.org/wiki/Théorème_des_deux_carrés_de_Fermat">Théorème des deux carrés de Fermat</a> (in French).
%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A009003 \ A004431.
%F A009003 INTERSECT A004439.
%e 13 is hypotenuse number A009003(3) but can be represented as A004431(3), so 13 is not in this sequence.
%t 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]; lst1={};Do[If[f[n^2]>0,AppendTo[lst1, n]],{n,3,5!}];lst1 (*A009003 Hypotenuse numbers (squares are sums of 2 distinct nonzero squares).*) lst2={};Do[If[f[n]>0,AppendTo[lst2, n]],{n,3,5!}];lst2 (*A004431 Numbers that are the sum of 2 distinct nonzero squares.*) Complement[lst1,lst2]
%Y Cf. A000404, A009003, A004431, A001481.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, Jul 07 2009
%E Formulas added, entries checked by _R. J. Mathar_, Aug 14 2009
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