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A058529
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Numbers whose prime factors are all congruent to +1 or -1 modulo 8.
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33
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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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
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MATHEMATICA
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Select[Range[500], Union[Abs[Mod[Transpose[FactorInteger[#]][[1]], 8, -1]]] == {1} &] (* T. D. Noe, Feb 07 2012 *)
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PROG
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(Haskell)
a058529 n = a058529_list !! (n-1)
a058529_list = filter (\x -> all (`elem` (takeWhile (<= x) a001132_list))
$ a027748_row x) [1..]
(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
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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William Bagby (bagsbee(AT)aol.com), Dec 24 2000
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EXTENSIONS
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Duplicated comment removed and name rewritten by Wolfdieter Lang, Feb 17 2015
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STATUS
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approved
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