

A009000


Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).


21



5, 10, 13, 15, 17, 20, 25, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 50, 51, 52, 53, 55, 58, 60, 61, 65, 65, 65, 65, 68, 70, 73, 74, 75, 75, 78, 80, 82, 85, 85, 85, 85, 87, 89, 90, 91, 95, 97, 100, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120
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OFFSET

1,1


COMMENTS

The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c.
If c^2 = a^2 + b^2 (a < b < c) then c^2 = (n^2 + m^2)/2 with n = b  a, m = b + a.  Zak Seidov, Mar 03 2011
Numbers n such that A083025(n) > 0, i.e., n is divisible by at least one prime of the form 4k+1.  Max Alekseyev, Oct 24 2008
A number appears only once in the sequence if and only if it is divisible by exactly one prime of the form 4k+1 with multiplicity one (cf. A084645).  JeanChristophe Hervé, Nov 11 2013
If c^2 = a^2 + b^2 with a and b > 0, then a <> b: the sum of 2 equal squares cannot be a square because sqrt(2) is not rational.  JeanChristophe Hervé, Nov 11 2013


REFERENCES

W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 8999 NCTM VA 1963.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108113 NCTM VA 1972.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 626 NCTM VA 1973.


LINKS

Zak Seidov and T. D. Noe, Table of n, a(n) for n = 1..10000 (Zak Seidov entered the first 1981 terms).
Anonymous, Links to Pythagorean Theorem Proofs
H. Bottomley, Pythagoras's theorem (animated proof)
Dept. of Pure Math., Univ. Sheffield, Animated Proof of Pythagoras Theorem [Broken link?]
T. Eveilleau, Animated proofs of the Pythagorean theorem:Sample Ancient Proofs (Text in French)
T. Eveilleau, More Animated proofs of the Pythagorean theorem (Text in French)
T. Eveilleau, An Experimental Illustration of the Pythagorean Theorem
Kangourou Math Website, L'animation du theoreme de Pythagore
Ron Knott, Pythagorean Triples and Online Calculators
Ron Knott, Rightangled Triangles and Pythagoras' Theorem
I. Kobayashi et al., Pythagorean Theorem(Java Interactive Proofs, Applications and Explorations)
Mathematical Database, Poster, 7 Ways to prove the Pythagorean Theorem
B. Richmond, The Pythagorean Theorem
M. Shepperd, Web Resources on Pythagoras' Theorem
J. S. Silverman, A Friendly Introduction to Number Theory, Chap.2:Pythagorean Triples; Chap.3:Pythagorean Triples and the Unit Circle
G. Villemin's Almanach of Numbers, Triangles & Triplets de Pythagore
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for sequences related to sums of squares


MATHEMATICA

max = 120; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[max], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; A009000 = Flatten[red /@ hypotenuses, 1][[All, 1]] (* JeanFrançois Alcover, May 23 2012, after Max Alekseyev *)


PROG

(PARI) list(lim)=my(v=List(), m2, s2, h2, h); for(middle=4, lim1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, h)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 23 2017


CROSSREFS

Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339.
Sequence in context: A226386 A200995 A049197 * A198389 A057100 A304436
Adjacent sequences: A008997 A008998 A008999 * A009001 A009002 A009003


KEYWORD

nonn,nice,easy


AUTHOR

David W. Wilson


STATUS

approved



