A084645 Hypotenuses for which there exists a unique integer-sided right triangle.

5, 10, 13, 15, 17, 20, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 51, 52, 53, 55, 58, 60, 61, 68, 70, 73, 74, 78, 80, 82, 87, 89, 90, 91, 95, 97, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120, 122, 123, 135, 136, 137, 140, 143, 146, 148, 149

Offset: 1

Author

Eric W. Weisstein, Jun 01 2003

Keywords

• nonn

Comments

Numbers whose square is uniquely decomposable into the sum of two nonzero squares: these are those numbers with exactly one prime divisor of the form 4k+1 with multiplicity one. - Jean-Christophe Hervé, Nov 11 2013

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Zak Seidov)
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A002144, A006339, A046080, A046109, A083025.

Cf. A004144 (0), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), A097101 (7), A290499 (8), A290500 (9), A097225 (10), A290501 (11), A097226 (12), A097102 (13), A290502 (14), A290503 (15), A097238 (16), A097239 (17), A290504 (18), A290505 (19), A097103 (22), A097244 (31), A097245 (37), A097282 (40), A097626 (67).

Programs

Mathematica

r[a_] := {b, c} /. {ToRules[ Reduce[0 < b < c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[ Range[150], Length[r[#]] == 1 &] (* Jean-François Alcover, Oct 22 2012 *)

PARI

is_a084645(n) = #qfbsolve(Qfb(1,0,1),n^2,3)==3 \\ Hugo Pfoertner, Sep 28 2024

Formula

Terms are obtained by the products A004144(k)*A002144(p) for k, p > 0, ordered by increasing values. - Jean-Christophe Hervé, Nov 12 2013

A046080(a(n)) = 1, A046109(a(n)) = 12. - Jean-Christophe Hervé, Dec 01 2013