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Search: id:A125667
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| A125667 |
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Eta numbers (from the Japanese word for "pariah" or "outcaste"). These are the positive odd integers which can not be used to make a hypotenuse of a primitive Pythagorean triangle (PPT). |
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+0
2
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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.
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REFERENCES
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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).
Frank Bernhart and H. Lee Price, Pythagorean Squares and Triangles, preprint (in preparation).
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FORMULA
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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
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EXAMPLE
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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
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CROSSREFS
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Cf. A020882.
Sequence in context: A027897 A027892 A047529 this_sequence A072939 A075607 A088630
Adjacent sequences: A125664 A125665 A125666 this_sequence A125668 A125669 A125670
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KEYWORD
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nonn
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AUTHOR
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H. Lee Price (tanutuva(AT)rochester.rr.com), Jan 29 2007, corrected Feb 03 2007
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