A125667 - OEIS

%I #14 Oct 22 2025 04:34:22

%S 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 69,71,75,77,79,81,83,87,91,93,95,99,103,105,107,111,115,117,119,121,

%U 123,127,129,131,133,135,139,141,143,147,151,153,155,159,161,163,165

%N 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 Eta numbers are the odd complement of A020882.

%C 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 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 H. Lee Price and Frank R. Bernhart, <a href="https://arxiv.org/abs/math/0701554">Pythagorean Triples and a New Pythagorean Theorem</a>, arXiv:math/0701554 [math.HO], 2007.

%H Frank Bernhart and H. Lee Price, <a href="https://arxiv.org/abs/math/0701624">Heron's Formula, Descartes Circles and Pythagorean Triangles</a>, arXiv:math/0701624 [math.MG], 2007.

%F 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 Class #1 a(6) = E because E is nonnegative, odd and not equal to (4k+1).

%e Class #2 a(4) = E because E is nonnegative, odd and E=(4k+1) but is not a member of A020882.

%t Select[Range[1, 300, 2], With[{p = PowersRepresentations[#^2, 2, 2]}, Length@p == 1 || Select[p, GCD @@ # == 1 &] == {}] &] (* _Oliver Seipel_, Oct 18 2025 *)

%Y Cf. A020882.

%K nonn

%O 1,2

%A _H. Lee Price_, Jan 29 2007, corrected Feb 03 2007