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A020882 Ordered hypotenuses (with multiplicity) of primitive Pythagorean triangles. 84
5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 65, 73, 85, 85, 89, 97, 101, 109, 113, 125, 137, 145, 145, 149, 157, 169, 173, 181, 185, 185, 193, 197, 205, 205, 221, 221, 229, 233, 241, 257, 265, 265, 269, 277, 281, 289, 293, 305, 305, 313, 317, 325, 325, 337, 349, 353, 365, 365 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The largest member 'c' of the primitive Pythagorean triples (a,b,c) ordered by increasing c.

These are numbers of the form a^2+b^2 where gcd( b-a, 2*a*b)=1. - M. F. Hasler, Apr 04 2010

Equivalently, numbers of the form a^2+b^2 where gcd(a,b) = 1 and a and b are not both odd. To avoid double-counting, require a > b > 0. - Franklin T. Adams-Watters, Mar 15 2015

The density of such points in a circle with radius squared = a(n) is ~ Pi * a(n). Restricting to a > b > 0 reduces this by a factor of 1/8; requiring gcd(a,b)=1 provides a factor of 6/Pi^2; and a, b not both odd is a factor of 2/3. (2/3, not 3/4, because the case a, b both even has already been eliminated.) Multiplying, a(n) * Pi * 1/8 * 6/Pi^2 * 2/3 is a(n) / (2 * Pi). But n is approximately this number of points, so a(n) ~ 2 * Pi * n. Conjectured by David W. Wilson, proof by Franklin T. Adams-Watters, Mar 15 2015

Permutations are in A094194, A088511, A121727, A119321, A113482 and A081804. Entries of A024409 occur here more than once. - R. J. Mathar, Apr 12 2010

As far as UNIQUE (ignoring repeating values) terms of A020882 are concerned - those seems to constitute a subset of the sequence, defined as: a(n)=(-1)^n+6*n for n>=1. - Alexander R. Povolotsky, Mar 15 2015

The terms in this sequence are given by f(m,n) = m^2 + n^2 where m and n are any two integers satisfying m > 1, n < m, the greatest common divisor of m and n is 1, and m and n are both not odd. E.g. f(m,n) = f(2,1) = 2^2 + 1^2 = 4 + 1 = 5. - Agola Kisira Odero, Apr 29 2016

REFERENCES

M. de Frénicle, "Méthode pour trouver la solutions des problèmes par les exclusions", in: "Divers ouvrages de mathématiques et de physique, par Messieurs de l'Académie royale des sciences", Paris, 1693, pp 1-44.

LINKS

M. F. Hasler and David W. Wilson, Table of n, a(n) for n = 1..10000, first 1593 terms from M. F. Hasler.

M. de Frénicle, Méthode pour trouver la solutions des problèmes par les exclusions B.N.F. permanent link to a scan of the original edition.

Werner Hürlimann, Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.5.

Hans Isdahl, Pythagoras site (in Norwegian)

Ron Knott, Pythagorean Triples and Online Calculators

H. P. Robinson and N. J. A. Sloane, Correspondence, 1971-1972

E. S. Rowland, Primitive Solutions to x^2 + y^2 = z^2

M. Somos, Table of primitive Pythagorean triplets and related parameters

Eric Weisstein's World of Mathematics, Pythagorean Triple

FORMULA

a(n) = sqrt((A120681(n)^2 + A120682(n)^2)/2). - Lekraj Beedassy, Jun 24 2006

a(n) = sqrt(A046086(n)^2 + A046087(n)^2). - Zak Seidov, Apr 12 2011

a(n) ~ 2*Pi*n. - observation by David W. Wilson, proved by Franklin T. Adams-Watters (cf. comments), Mar 15 2015

MATHEMATICA

t={}; Do[Do[a=Sqrt[c^2-b^2]; If[a>b, Break[]]; If[IntegerQ[a]&&GCD[a, b, c]==1, AppendTo[t, c]], {b, c-1, 3, -1}], {c, 400}]; t (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)

f[c_] := Block[{a = 1, b, lst = {}}, While[b = Sqrt[c^2 - a^2]; a < b, If[ IntegerQ@ b && GCD[a, b, c] == 1, AppendTo[lst, a]]; a++]; lst]

Select[1 + 4 Range@ 100, Length@ f@# > 0 &] (* Robert G. Wilson v, Mar 16 2014 *)

PROG

(PARI) {my( c=0, new=[]); for( b=1, 99, for( a=1, b-1, gcd(b-a, 2*a*b) == 1 && new=concat(new, a^2+b^2)); new=vecsort(new); for( j=1, #new, new[j] > (b+1)^2 & (new=vecextract(new, Str(j, ".."))) & next(2); write("b020882.txt", c++, " "new[j])); new=[])} \\ M. F. Hasler, Apr 04 2010

CROSSREFS

Cf. A004613, A008846, A020883-A020886, A046086, A046087, A134961, A222946 (as a number triangle).

Sequence in context: A037046 A126887 A087445 * A081804 A004613 A008846

Adjacent sequences:  A020879 A020880 A020881 * A020883 A020884 A020885

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Edited by N. J. A. Sloane, May 15 2010

STATUS

approved

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Last modified August 21 23:50 EDT 2017. Contains 290941 sequences.