|
| |
|
|
A058529
|
|
Numbers whose prime factors are all == +/-1 (mod 8).
|
|
25
| |
|
|
1, 7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 127, 137, 151, 161, 167, 191, 193, 199, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 289, 311, 313, 329, 337, 343, 353, 359, 367, 383, 391, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Numbers of the form x^2 - 2*y^2, where x is odd and x and y are relatively prime. - Franklin T. Adams-Watters, Jun 24 2011.
Consider primitive Pythagorean triangles (a^2 + b^2 = c^2, gcd(a, b) = 1, a <= b); sequence gives values b-a, sorted with duplicates removed; terms > 1 in sequence give values of a + b, sorted. (See A046086 and A046087.)
Ordered set of (semiperimeter + radius of largest inscribed circle) of all primitive Pythagorean triangles. Semiperimeter of Pythagorean triangle + radius of largest circle inscribed in triangle = ((a+b+c)/2) + ((a+b-c)/2) = a + b.
(a + b) for all primitive Pythagorean triples (a, b, c), sorted (see A046086 and A046087).
The terms of this sequence are all of the form 6*N+/-1, since the prime divisors are, and numbers of this form are closed under multiplication. In fact, all terms are == 1, 7, 17, or 23 (mod 24). [From J.T.Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 28 2009], edited by Franklin T. Adams-Watters, Jun 24 2011.
Is similar to A001132, but includes composites whose factors are in A001132. Can be generated in this manner.
|
|
|
REFERENCES
| a(n+1) in Frenicle page 31: Methode pour trouver .., 44 pages.In Divers ouvrages de mathematique .. Par Messieurs de l'Academie Royale des Sciences, in-fol, 6+518+1PP, Paris, 1693. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 06 2008]
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=1..1000
K. S. Brown, Pythagorean graphs
F. Barnes, primitive Pythagorean triangles where a-b is a constant
|
|
|
MATHEMATICA
| Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)
|
|
|
CROSSREFS
| Cf. A020882-A020886, A020888, A046086, A046087, A014498, A001132, A001653.
Sequence in context: A032454 A107643 A198441 * A120681 A001132 A165353
Adjacent sequences: A058526 A058527 A058528 * A058530 A058531 A058532
|
|
|
KEYWORD
| easy,nice,nonn,changed
|
|
|
AUTHOR
| William Bagby (bagsbee(AT)aol.com), Dec 24 2000
|
|
|
EXTENSIONS
| More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jul 02 2001
Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jun 24 2011
|
| |
|
|
