A097101 Numbers n that are the hypotenuse of exactly 7 distinct integer-sided right triangles, i.e., n^2 can be written as a sum of two squares in 7 ways.

325, 425, 650, 725, 845, 850, 925, 975, 1025, 1275, 1300, 1325, 1445, 1450, 1525, 1690, 1700, 1825, 1850, 1950, 2050, 2175, 2225, 2275, 2425, 2525, 2535, 2550, 2600, 2650, 2725, 2775, 2825, 2873, 2890, 2900, 2925, 2975

Offset: 1

Author

James R. Buddenhagen, Sep 15 2004

Keywords

• nonn

Comments

Comment from R. J. Mathar, Feb 26 2008, edited by Zak Seidov May 12 2008: (Start)

There are nonsquares x which can be written as a sum of 2 nonzero squares in exactly 7 different ways and which are by definition not in this sequence.

203125 = (125*sqrt(13))^2 is the first example: 203125 = 625 + 202500 = 10404 + 192721 = 18225 + 184900= 22500 + 180625= 62500 + 140625= 69169 + 133956= 84100 + 119025.

The second and third examples are 265625 = (125*sqrt(17))^2 and 406250=(125*sqrt(26))^2. (End)

If m is a term, then 2*m and p*m are terms where p is any prime of the form 4k+3. - Ray Chandler, Dec 30 2019

Examples

			Example supplied by _R. J. Mathar_, Feb 26 2008:
The smallest number that can be written as a sum of two nonzero squares in 7 different ways is 105625 = 325^2:
1296 + 104329 = 105625 = 325^2
6400 + 99225 = 105625 = 325^2
8281 + 97344 = 105625 = 325^2
15625 + 90000 = 105625 = 325^2
27225 + 78400 = 105625 = 325^2
38025 + 67600 = 105625 = 325^2
41616 + 64009 = 105625 = 325^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004144 (0), A084645 (1), A084646 (2), A084647 (3), A084648 (4), A084649 (5), A097219 (6), 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[0Vincenzo Librandi, Mar 01 2016 *)

Formula

Equals {n: A025426(n^2)=7}.

Extensions

Definition and comments corrected by Zak Seidov, Feb 26 2008, May 12 2008