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).

1, 3, 7, 9, 11, 15, 19, 21, 23, 27, 31, 33, 35, 39, 43, 45, 47, 49, 51, 55, 57, 59, 63, 67, 69, 71, 75, 77, 79, 81, 83, 87, 91, 93, 95, 99, 103, 105, 107, 111, 115, 117, 119, 121, 123, 127, 129, 131, 133, 135, 139, 141, 143, 147, 151, 153, 155, 159, 161, 163, 165

Offset

1,2

Comments

Eta numbers are the odd complement of A020882.

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.

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.

Links

Table of n, a(n) for n=1..61.

H. Lee Price and Frank R. Bernhart, Pythagorean Triples and a New Pythagorean Theorem, arXiv:math/0701554 [math.HO], 2007.

Frank Bernhart and H. Lee Price, Heron's Formula, Descartes Circles and Pythagorean Triangles, arXiv:math/0701624 [math.MG], 2007.

Formula

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.

Example

Class #1 a(6) = E because E is nonnegative, odd and not equal to (4k+1).
Class #2 a(4) = E because E is nonnegative, odd and E=(4k+1) but is not a member of A020882.

Mathematica

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

Crossrefs

Cf. A020882.

Sequence in context: A158938 A047529 A359567 * A072939 A171947 A287914

Adjacent sequences: A125664 A125665 A125666 * A125668 A125669 A125670

Keyword

nonn

Author

H. Lee Price, Jan 29 2007, corrected Feb 03 2007

Status

approved