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A125667 Eta numbers (from the Japanese word for "pariah" or "outcast"). These are the positive odd integers which cannot be used to make a hypotenuse of a primitive Pythagorean triangle (PPT).

%I A125667 #9 Dec 07 2015 00:33:08
%S A125667 1,3,7,9,11,15,19,21,23,27,31,33,35,39,43,45,47,49,51,55,57,59,63,67,
%T A125667 69,71,75,77,79,81,83,87,91,93,95,99,103,105,107,111,115,117,119,121,
%U A125667 123,127,129,131,133,135,139,141,143,147,151,153,155,159,161,163,165
%N A125667 Eta numbers (from the Japanese word for "pariah" or "outcast"). These are the positive odd integers which cannot be used to make a hypotenuse of a primitive Pythagorean triangle (PPT).
%C A125667 Eta numbers are the odd complement of A020882.
%C A125667 Properties: A PPT hypotenuse has form (4k+1), but the converse is not true. Thus Eta numbers fall into two classes: #1 Odd integers which do not have form (4k+1), #2 Odd integers of form (4k+1) which are not members of A020882.
%C A125667 Eta numbers >1 can be the leg of PPT[a,b,c] but not a hypotenuse, while members of A020882 can be both. By Fermat's theorem, class #2 eta numbers are not prime.
%H A125667 H. Lee Price and Frank R. Bernhart, <a href="http://arxiv.org/abs/math.HO/0701554">Pythagorean Triples and a New Pythagorean Theorem</a>, arXiv:math.HO/0701554, (2007).
%H A125667 Frank Bernhart and H. Lee Price, <a href="http://arxiv.org/abs/math.MG/0701624">Heron's Formula, Descartes Circles and Pythagorean Triangles</a>, arXiv:math.MG/0701624, (2007).
%F A125667 Class #1 a(n) = E because E is nonnegative, odd and not equal to (4k+1). Class #2 a(n) = E because E=(4k+1) (not class #1) but is not a member of A020882.
%e A125667 Class #1 a(6) = E because E is nonnegative, odd and not equal to (4k+1).
%e A125667 Class #2 a(4) = E because E is nonnegative, odd and E=(4k+1) but is not a member of A020882.
%Y A125667 Cf. A020882.
%K A125667 nonn
%O A125667 1,2
%A A125667 _H. Lee Price_, Jan 29 2007, corrected Feb 03 2007