A162592 Hypotenuse numbers A009003 which cannot be represented as sum of 2 distinct nonzero squares.

15, 30, 35, 39, 51, 55, 60, 70, 75, 78, 87, 91, 95, 102, 105, 110, 111, 115, 119, 120, 123, 135, 140, 143, 150, 155, 156, 159, 165, 174, 175, 182, 183, 187, 190, 195, 203, 204, 210, 215, 219, 220, 222, 230, 235, 238, 240, 246, 247, 255, 259, 267, 270, 273, 275

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

Keywords

• nonn

Comments

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

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

Examples

			13 is hypotenuse number A009003(3) but can be represented as A004431(3), so 13 is not in this sequence.
		

Links

• Eric Weisstein's World of Mathematics, Square Number
• Proof Wiki, Fermat's Christmas Theorem
• Wikipedia, Fermat's Theorem on sums of two squares
• Wikipedia (fr), Théorème des deux carrés de Fermat (in French).
• Index entries for sequences related to sums of squares

Crossrefs

Cf. A000404, A009003, A004431, A001481.

Programs

Mathematica

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]

Formula

A009003 \ A004431.

A009003 INTERSECT A004439.

Extensions

Formulas added, entries checked by R. J. Mathar, Aug 14 2009