

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



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
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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.HO/0701554, (2007).
Frank Bernhart and H. Lee Price, Heron's Formula, Descartes Circles and Pythagorean Triangles, arXiv:math.MG/0701624, (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.


CROSSREFS

Cf. A020882.
Sequence in context: A027892 A158938 A047529 * 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



