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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.

%I A097101 #19 Dec 30 2019 10:14:00
%S A097101 325,425,650,725,845,850,925,975,1025,1275,1300,1325,1445,1450,1525,
%T A097101 1690,1700,1825,1850,1950,2050,2175,2225,2275,2425,2525,2535,2550,
%U A097101 2600,2650,2725,2775,2825,2873,2890,2900,2925,2975
%N 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.
%C A097101 Comment from _R. J. Mathar_, Feb 26 2008, edited by _Zak Seidov_ May 12 2008: (Start)
%C A097101 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.
%C A097101 203125 = (125*sqrt(13))^2 is the first example: 203125 = 625 + 202500 = 10404 + 192721 = 18225 + 184900= 22500 + 180625= 62500 + 140625= 69169 + 133956= 84100 + 119025.
%C A097101 The second and third examples are 265625 = (125*sqrt(17))^2 and 406250=(125*sqrt(26))^2. (End)
%C A097101 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
%H A097101 Chai Wah Wu, <a href="/A097101/b097101.txt">Table of n, a(n) for n = 1..10000</a>
%F A097101 Equals {n: A025426(n^2)=7}.
%e A097101 Example supplied by _R. J. Mathar_, Feb 26 2008:
%e A097101 The smallest number that can be written as a sum of two nonzero squares in 7 different ways is 105625 = 325^2:
%e A097101 1296 + 104329 = 105625 = 325^2
%e A097101 6400 + 99225 = 105625 = 325^2
%e A097101 8281 + 97344 = 105625 = 325^2
%e A097101 15625 + 90000 = 105625 = 325^2
%e A097101 27225 + 78400 = 105625 = 325^2
%e A097101 38025 + 67600 = 105625 = 325^2
%e A097101 41616 + 64009 = 105625 = 325^2.
%t A097101 r[a_]:={b,c}/.{ToRules[Reduce[0<b<c && a^2 == b^2 + c^2, {b, c}, Integers]]}; Select[Range[3000], Length[r[#]] == 7 &] (* _Vincenzo Librandi_, Mar 01 2016 *)
%Y A097101 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).
%K A097101 nonn
%O A097101 1,1
%A A097101 _James R. Buddenhagen_, Sep 15 2004
%E A097101 Definition and comments corrected by _Zak Seidov_, Feb 26 2008, May 12 2008