login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A058529 Numbers whose prime factors are all congruent to +1 or -1 modulo 8. 31
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; text; 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.

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

REFERENCES

Olaf Delgado-Friedrichs and Michael O’Keeffe, Edge-transitive lattice nets, Acta Cryst. (2009). A65, 360-363.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

F. Barnes, primitive Pythagorean triangles where a-b is a constant

K. S. Brown, Pythagorean graphs

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

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

CROSSREFS

Cf. A020882-A020886, A020888, A046086, A046087, A014498, A001132, A001653, A027748, A047522.

Sequence in context: A319040 A216838 A198441 * A253408 A120681 A270951

Adjacent sequences:  A058526 A058527 A058528 * A058530 A058531 A058532

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 25 03:50 EDT 2019. Contains 321450 sequences. (Running on oeis4.)