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A162592
Hypotenuse numbers A009003 which cannot be represented as sum of 2 distinct nonzero squares.
1
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
1,1
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
FORMULA
A009003 INTERSECT A004439.
EXAMPLE
13 is hypotenuse number A009003(3) but can be represented as A004431(3), so 13 is not in this sequence.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Formulas added, entries checked by R. J. Mathar, Aug 14 2009
STATUS
approved