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A058529
Numbers whose prime factors are all congruent to +1 or -1 modulo 8.
33
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
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.
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). - 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.
Third side of primitive parallepipeds with square base; that is, integer solution of a^2 + b^2 + c^2 = d^2 with gcd(a,b,c) = 1 and b = c. - Carmine Suriano, May 03 2013
Other than -1, values of difference z-y that solve the Diophantine equation x^2 + y^2 = z^2 + 2. - Carmine Suriano, Jan 05 2015
For k > 1, k is in the sequence iff A330174(k) > 0. - Ray Chandler, Feb 26 2020
REFERENCES
B Berggren, Pytagoreiska trianglar. Tidskrift för elementär matematik, fysik och kemi, 17:129-139, 1934.
Olaf Delgado-Friedrichs and Michael O’Keeffe, Edge-transitive lattice nets, Acta Cryst. (2009). A65, 360-363.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
K. S. Brown, Pythagorean graphs.
O. Delgado-Friedrichs and M. O'Keeffe, Edge-transitive lattice nets, Acta Cryst. A, A65 (2009), 360-363.
B. Frénicle, Méthode pour trouver la solution des problèmes par les exclusions, 44 pages (see p. 31). In Divers ouvrages de mathematique .. Par Messieurs de l'Academie Royale des Sciences, in-fol, 6+518+1PP, Paris, 1693. - Paul Curtz, Sep 06 2008
FORMULA
a(n) = |A-B|=|j^2-2*k^2|, j=(2*n-1), k,n in N, GCD(j,k)=1, the absolute difference between primitive Pythagorean triple legs (sides adjacent to the right angle). - Roger M Ellingson, Dec 09 2023
MATHEMATICA
Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)
PROG
(Haskell)
a058529 n = a058529_list !! (n-1)
a058529_list = filter (\x -> all (`elem` (takeWhile (<= x) a001132_list))
$ a027748_row x) [1..]
-- Reinhard Zumkeller, Jan 29 2013
(PARI) is(n)=my(f=factor(n)[, 1]%8); for(i=1, #f, if(f[i]!=1 && f[i]!=7, return(0))); 1 \\ Charles R Greathouse IV, Aug 01 2016
KEYWORD
easy,nice,nonn
AUTHOR
William Bagby (bagsbee(AT)aol.com), Dec 24 2000
EXTENSIONS
More terms from Naohiro Nomoto, Jul 02 2001
Edited by Franklin T. Adams-Watters, Jun 24 2011
Duplicated comment removed and name rewritten by Wolfdieter Lang, Feb 17 2015
STATUS
approved