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 A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares). 35
 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 (list; graph; refs; listen; history; text; internal format)
 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). - Jean-Christophe 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. - Jean-Christophe Hervé, Nov 11 2013 REFERENCES W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 89-99 NCTM VA 1963. W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108-113 NCTM VA 1972. W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 62-6 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) [broken link] T. Eveilleau, An Experimental Illustration of the Pythagorean Theorem, (requires a flash player) Kangourou Math Website, L'animation du theoreme de Pythagore Ron Knott, Pythagorean Triples and Online Calculators Ron Knott, Right-angled 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, Chapters 1 to 6 (see Chapters 2 and 3). 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]] (* Jean-François Alcover, May 23 2012, after Max Alekseyev *) PROG (PARI) list(lim)=my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, 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 (PARI) list(lim) = {my(lh = List()); for(u = 2, sqrtint(lim), for(v = 1, u, if (u^2+v^2 > lim, break); if ((gcd(u, v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u^2+v^2) > lim, break); /* if (u^2 - v^2 < 2*u*v, w = [i*(u^2 - v^2), i*2*u*v, i*(u^2+v^2)], w = [i*2*u*v, i*(u^2 - v^2), i*(u^2+v^2)]); */ listput(lh, i*(u^2+v^2)); ); ); ); ); vecsort(Vec(lh)); } \\ Michel Marcus, Apr 10 2021 (Python) from math import isqrt def aupto(limit): s = [i*i for i in range(1, limit+1)] s2 = sorted(a+b for i, a in enumerate(s) for b in s[i+1:]) return [isqrt(k) for k in s2 if k in s] print(aupto(120)) # Michael S. Branicky, May 10 2021 CROSSREFS Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339. Sequence in context: A200995 A049197 A324928 * A198389 A057100 A304436 Adjacent sequences: A008997 A008998 A008999 * A009001 A009002 A009003 KEYWORD nonn,nice,easy AUTHOR David W. Wilson STATUS approved

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Last modified September 28 02:24 EDT 2023. Contains 365714 sequences. (Running on oeis4.)